Accuracy Verification: Calculator Results Checked Against SciPy

T-Test Calculator

Welch two-sample t-test · reference: `scipy.stats.ttest_ind(a, b, equal_var=False)`
Quantity	Reference value	Our result	Status
t statistic	-3.92195	-3.9219	✓ matches
p-value	0.001534143	0.0015341	✓ matches
Degrees of freedom	13.99955	14	✓ matches

Pooled two-sample t-test · reference: `scipy.stats.ttest_ind(a, b, equal_var=True)`
Quantity	Reference value	Our result	Status
t statistic	-3.92195	-3.9219	✓ matches
p-value	0.001534056	0.0015341	✓ matches
Degrees of freedom	14	14	✓ matches

Paired t-test (after − before) · reference: `scipy.stats.ttest_rel(after, before)`
Quantity	Reference value	Our result	Status
t statistic	26.93836	26.938	✓ matches
p-value	2.490034e-8	< 0.0001	✓ matches
Degrees of freedom	7	7	✓ matches

One-sample t-test against μ₀ = 24 · reference: `scipy.stats.ttest_1samp(a, 24)`
Quantity	Reference value	Our result	Status
t statistic	1.550492	1.5505	✓ matches
p-value	0.1649589	0.16496	✓ matches
Degrees of freedom	7	7	✓ matches

One-sample t-test, left-tailed · reference: `scipy.stats.ttest_1samp(a, 24, alternative='less')`
Quantity	Reference value	Our result	Status
p-value	0.9175205	0.91752	✓ matches

One-Sample Z-Test Calculator

One-sample z-test, two-tailed · reference: `2 × scipy.stats.norm.sf(|z|)`
Quantity	Reference value	Our result	Status
z-statistic	2	2	✓ matches
p-value	0.04550026	0.0455	✓ matches
Standard error (σ/√n)	2.5	2.5	✓ matches

One-sample z-test, left-tailed · reference: `scipy.stats.norm.cdf(z)`
Quantity	Reference value	Our result	Status
z-statistic	-1.5	-1.5	✓ matches
p-value	0.0668072	0.066807	✓ matches

One-Way ANOVA Calculator

One-way ANOVA, 3 groups · reference: `scipy.stats.f_oneway(g1, g2, g3)`
Quantity	Reference value	Our result	Status
F statistic	12.63636	12.636	✓ matches
p-value	0.0006068783	0.00060688	✓ matches

One-way ANOVA, 4 groups · reference: `scipy.stats.f_oneway(*groups)`
Quantity	Reference value	Our result	Status
F statistic	43.10061	43.101	✓ matches
p-value	6.854293e-8	< 0.0001	✓ matches

Chi-Square Test Calculator

Chi-square test of independence (2 × 3) · reference: `scipy.stats.chi2_contingency(table, correction=False)`
Quantity	Reference value	Our result	Status
Chi-square statistic χ²	9.863636	9.86364	✓ matches
p-value	0.007213376	0.0072134	✓ matches
Degrees of freedom	2	2	✓ matches
Cramér’s V (effect size)	0.2654328	0.2654	✓ matches

Chi-square goodness of fit (equal expected) · reference: `scipy.stats.chisquare(observed)`
Quantity	Reference value	Our result	Status
Chi-square statistic χ²	12.32	12.32	✓ matches
p-value	0.00636363	0.0063636	✓ matches

Chi-square goodness of fit (given expected) · reference: `scipy.stats.chisquare(observed, f_exp)`
Quantity	Reference value	Our result	Status
Chi-square statistic χ²	2.666667	2.66667	✓ matches
p-value	0.4459217	0.44592	✓ matches

P-Value from Chi-Square Calculator

χ² = 7.81, df = 3 · reference: `scipy.stats.chi2.sf / ppf`
Quantity	Reference value	Our result	Status
p-value (right-tail)	0.05010606	0.0501061	✓ matches
Critical value (right tail, α = 0.05)	7.814728	7.81473	✓ matches

χ² = 18.307, df = 10 · reference: `scipy.stats.chi2.sf / ppf`
Quantity	Reference value	Our result	Status
p-value (right-tail)	0.05000059	0.0500006	✓ matches
Critical value (right tail, α = 0.05)	18.30704	18.307	✓ matches

χ² = 1.2, df = 2 · reference: `scipy.stats.chi2.sf / ppf`
Quantity	Reference value	Our result	Status
p-value (right-tail)	0.5488116	0.548812	✓ matches
Critical value (right tail, α = 0.05)	5.991465	5.99146	✓ matches

Regression Equation Calculator

Simple linear regression · reference: `scipy.stats.linregress(x, y)`
Quantity	Reference value	Our result	Status
Slope b₁	1.978571	1.97857	✓ matches
Intercept b₀	0.1714286	0.171429	✓ matches
Correlation r	0.9976494	0.99765	✓ matches
R² (coefficient of determination)	0.9953043	0.9953	✓ matches
Prediction at x = 10	19.95714	19.9571	✓ matches

Weak relationship (non-significant slope) · reference: `scipy.stats.linregress(x, y)`
Quantity	Reference value	Our result	Status
Slope b₁	0.1826008	0.182601	✓ matches
Intercept b₀	29.72406	29.7241	✓ matches
Correlation r	0.3952696	0.39527	✓ matches
p-value for the slope	0.3324464	0.33245	✓ matches

Confidence Interval Calculator

Confidence interval for a mean (t, raw data) · reference: `scipy.stats.t.interval(0.95, n − 1, mean, sem)`
Quantity	Reference value	Our result	Status
95% confidence interval for the mean	12.15685 to 13.09315	12.1568 to 13.0932	✓ matches
Sample mean	12.625	12.625	✓ matches

Wilson interval for a proportion (45 of 120) · reference: `scipy.stats.binomtest(45, 120).proportion_ci(method='wilson')`
Quantity	Reference value	Our result	Status
95% CI (Wilson, recommended)	29.35243 to 46.42305	29.352 % to 46.423 %	✓ matches

Mean and Standard Deviation Calculator

Sample mean and SD · reference: `numpy.std(ddof=1), numpy.var(ddof=1)`
Quantity	Reference value	Our result	Status
Sample standard deviation (s)	2.13809	2.13809	✓ matches
Mean (x̄)	5	5	✓ matches
Sample variance (s²)	4.571429	4.57143	✓ matches

Population SD · reference: `numpy.std(ddof=0)`
Quantity	Reference value	Our result	Status
Population standard deviation (σ)	2	2	✓ matches
Population variance (σ²)	4	4	✓ matches

Mean Absolute Deviation (MAD) Calculator

Mean absolute deviation · reference: `numpy.mean(|x − mean|), scipy.stats.median_abs_deviation`
Quantity	Reference value	Our result	Status
Mean absolute deviation (MAD)	3.75	3.75	✓ matches
Median absolute deviation	2.5	2.5	✓ matches

RSD Calculator: Relative Standard Deviation

Relative standard deviation · reference: `numpy.std(ddof=1) / numpy.mean × 100`
Quantity	Reference value	Our result	Status
Relative standard deviation (RSD)	2.368519	2.3685 %	✓ matches

Mean, Median, Mode and Range Calculator

Mean, median, range · reference: `numpy.mean, numpy.median`
Quantity	Reference value	Our result	Status
Mean	6.125	6.125	✓ matches
Median	7	7	✓ matches
Range	8	8	✓ matches

Skewness Calculator

Skewness (types 1 and 2) · reference: `scipy.stats.skew(bias=False / True)`
Quantity	Reference value	Our result	Status
G₁: adjusted Fisher-Pearson (type 2, Excel SKEW)	1.543273	1.5433	✓ matches
g₁: moment coefficient (type 1)	1.343247	1.3432	✓ matches

Kurtosis Calculator

Excess kurtosis (types 1 and 2) · reference: `scipy.stats.kurtosis(bias=False / True)`
Quantity	Reference value	Our result	Status
G₂: adjusted (type 2, Excel KURT)	1.642372	1.6424	✓ matches
g₂: moment coefficient (type 1)	0.5721223	0.57212	✓ matches

Standard Error of the Mean (SEM) Calculator

Standard error of the mean · reference: `scipy.stats.sem`
Quantity	Reference value	Our result	Status
Standard error of the mean (SEM)	0.9689043	0.968904	✓ matches
Mean (x̄)	14.71429	14.7143	✓ matches

Sample Variance Calculator

Sample and population variance · reference: `numpy.var(ddof=1 / 0)`
Quantity	Reference value	Our result	Status
Sample variance (s²)	6.571429	6.571429	✓ matches
Population variance (σ²)	5.632653	5.632653	✓ matches

Binomial Probability Calculator with Steps

Binomial n = 10, p = 0.4, k = 3 · reference: `scipy.stats.binom.pmf / cdf / sf`
Quantity	Reference value	Our result	Status
P(X = 3)	0.2149908	0.21499085	✓ matches
P(X ≤ 3)	0.3822806	0.3822806	✓ matches
P(X ≥ 3)	0.8327102	0.83271025	✓ matches
P(X > 3)	0.6177194	0.6177194	✓ matches

Binomial n = 50, p = 0.07, k = 5 · reference: `scipy.stats.binom.pmf / cdf`
Quantity	Reference value	Our result	Status
P(X = 5)	0.1359267	0.13592666	✓ matches
P(X ≤ 5)	0.8649536	0.86495357	✓ matches

Geometric Distribution Calculator

Geometric p = 0.2, k = 4 · reference: `scipy.stats.geom.pmf / cdf`
Quantity	Reference value	Our result	Status
P(X = 4)	0.1024	0.1024	✓ matches
P(success within 4 trials)	0.5904	0.5904	✓ matches
Expected number of trials 1/p	5	5	✓ matches

Gamma Distribution Calculator

Gamma shape 3, scale 2, x = 5 · reference: `scipy.stats.gamma.cdf / pdf`
Quantity	Reference value	Our result	Status
P(X ≤ 5)	0.4561869	0.45618688	✓ matches
Density f(5)	0.1282578	0.12825781	✓ matches
Mean kθ	6	6	✓ matches
Variance kθ²	12	12	✓ matches

Z-Table Calculator

Standard normal at z = 1.96 · reference: `scipy.stats.norm.cdf / sf`
Quantity	Reference value	Our result	Status
Area to the left P(Z < z)	0.9750021	0.9750021	✓ matches
Area to the right P(Z > z)	0.0249979	0.0249979	✓ matches

Standard normal at z = -0.5 · reference: `scipy.stats.norm.cdf / sf`
Quantity	Reference value	Our result	Status
Area to the left P(Z < z)	0.3085375	0.3085375	✓ matches
Area to the right P(Z > z)	0.6914625	0.6914625	✓ matches

Standard normal at z = 2.576 · reference: `scipy.stats.norm.cdf / sf`
Quantity	Reference value	Our result	Status
Area to the left P(Z < z)	0.9950025	0.9950025	✓ matches
Area to the right P(Z > z)	0.004997532	0.004997532	✓ matches

Z Test Statistic Calculator

z for a sample mean · reference: `scipy.stats.norm.sf`
Quantity	Reference value	Our result	Status
z for a sample mean	1.725	1.725	✓ matches
Two-tailed p-value	0.08452747	0.084527	✓ matches

nCr Calculator (Combinations)

C(52, 5) · reference: `math.comb(52, 5)`
Quantity	Reference value	Our result	Status
C(52, 5) = 52C5	2598960	2,598,960	✓ matches

C(40, 20) exact integer · reference: `math.comb(40, 20)`
Quantity	Reference value	Our result	Status
C(40, 20) = 40C20	137846528820	137,846,528,820	✓ matches

C(100, 50) in scientific notation · reference: `math.comb(100, 50)`
Quantity	Reference value	Our result	Status
C(100, 50) = 100C50	1.008913e+29	1.00891 × 10^29 (30 digits)	✓ matches

nPr Calculator (Permutations)

P(10, 4) · reference: `math.perm(10, 4)`
Quantity	Reference value	Our result	Status
P(10, 4) = 10P4	5040	5,040	✓ matches

Combination Calculator (nCr and nPr)

C(49, 6) lottery · reference: `math.comb, math.perm`
Quantity	Reference value	Our result	Status
Combinations C(n, r)	13983816	13,983,816	✓ matches
Permutations P(n, r)	10068347520	10,068,347,520	✓ matches

Birthday Paradox Calculator

Birthday paradox, 23 people · closed form: `exact product 1 − ∏(365 − i)/365`
Quantity	Reference value	Our result	Status
Chance at least two of 23 share a birthday	0.5072972	0.5072972	✓ matches

Mean Squared Error (MSE) Calculator

MSE, RMSE, MAE · reference: `numpy`
Quantity	Reference value	Our result	Status
Mean squared error (MSE)	1.071429	1.071429	✓ matches
Root mean squared error (RMSE)	1.035098	1.0351	✓ matches
Mean absolute error (MAE)	1	1	✓ matches

SSE Calculator (Sum of Squared Errors)

Sum of squared errors · reference: `numpy`
Quantity	Reference value	Our result	Status
Sum of squared errors (SSE)	0.59	0.59	✓ matches
Total sum of squares (SST)	80.83333	80.8333	✓ matches

Cosine Similarity Calculator

Cosine similarity · reference: `numpy.dot, numpy.linalg.norm`
Quantity	Reference value	Our result	Status
Cosine similarity	0.9722843	0.9722843	✓ matches
Dot product A·B	2557	2557	✓ matches
Euclidean distance	16.94107	16.9411	✓ matches

Entropy Calculator (Shannon Entropy)

Shannon entropy of counts 9, 5 (bits) · reference: `scipy.stats.entropy(base=2)`
Quantity	Reference value	Our result	Status
Shannon entropy H (bits)	0.940286	0.940286	✓ matches

F1 Score Calculator

F1 from confusion counts · closed form: `F1 = 2TP / (2TP + FP + FN)`
Quantity	Reference value	Our result	Status
F1 score	0.75	0.75	✓ matches
Precision	80	80 %	✓ matches
Recall	70.58824	70.588 %	✓ matches

Doubling Time Calculator

Doubling time at 3.5% growth · closed form: `ln 2 / ln(1.035)`
Quantity	Reference value	Our result	Status
Doubling time (years)	20.14879	20.1488	✓ matches

Cochran’s Formula Calculator

Cochran sample size (95%, ±5%, p = 0.5, N = 2000) · closed form: `n₀ = z²pq/e², n = n₀ / (1 + (n₀ − 1)/N)`
Quantity	Reference value	Our result	Status
Cochran’s n₀ (infinite population)	385	385	✓ matches
After finite population correction	323	323	✓ matches

Probability of 3 Events Calculator

3 independent events (exact enumeration of all outcomes) · closed form: `enumeration of all 2ⁿ outcomes with independent probabilities`
Quantity	Reference value	Our result	Status
All 3 events occur P(A ∩ B ∩ C)	0.03	0.03	✓ matches
None occur	0.28	0.28	✓ matches
At least one occurs P(A ∪ B ∪ …)	0.72	0.72	✓ matches
Exactly 1 of 3 occur	0.47	0.47	✓ matches
Exactly 2 of 3 occur	0.22	0.22	✓ matches
Expected number of events	1	1	✓ matches
At least 2 occur	0.25	0.25	✓ matches

4 Event Probability Calculator

4 independent events (exact enumeration of all outcomes) · closed form: `enumeration of all 2ⁿ outcomes with independent probabilities`
Quantity	Reference value	Our result	Status
All 4 events occur P(A ∩ B ∩ C ∩ D)	0.009	0.009	✓ matches
None occur	0.126	0.126	✓ matches
At least one occurs P(A ∪ B ∪ …)	0.874	0.874	✓ matches
Exactly 1 of 4 occur	0.383	0.383	✓ matches
Exactly 2 of 4 occur	0.365	0.365	✓ matches
Expected number of events	1.5	1.5	✓ matches
At least 2 occur	0.491	0.491	✓ matches

Binomial Distribution Table Calculator

Binomial n = 12, p = 0.35 summary · reference: `scipy.stats.binom(12, 0.35).mean / std / var / stats("s") / median`
Quantity	Reference value	Our result	Status
Mean μ = np	4.2	4.2	✓ matches
Standard deviation σ	1.652271	1.65227	✓ matches
Variance	2.73	2.73	✓ matches
Skewness	0.1815683	0.1816	✓ matches
Median	4	4	✓ matches

Combinations with Repetition Calculator

Multisets: 8 items from 5 types · reference: `math.comb(n + r − 1, r), n ** r, math.comb(r − 1, n − 1)`
Quantity	Reference value	Our result	Status
Combinations with repetition, 8 from 5 types	495	495	✓ matches
Ordered with repetition nʳ	390625	390,625	✓ matches
Positive integer solutions (each xᵢ ≥ 1)	35	35	✓ matches

Confusion Matrix Calculator

Confusion matrix metrics · closed form: `standard definitions (scikit-learn conventions)`
Quantity	Reference value	Our result	Status
Accuracy	87.5	87.5 %	✓ matches
Precision (PPV)	89.47368	89.47 %	✓ matches
Recall (sensitivity, TPR)	85	85 %	✓ matches
Specificity (TNR)	90	90 %	✓ matches
F1 score	87.17949	87.18 %	✓ matches
Negative predictive value	85.71429	85.71 %	✓ matches
False positive rate	10	10 %	✓ matches
False negative rate (miss rate)	15	15 %	✓ matches

Hypothesis Testing Calculator

One-sample t-test from summary statistics · reference: `scipy.stats.t.sf`
Quantity	Reference value	Our result	Status
Test statistic t	1.844662	1.8447	✓ matches
p-value	0.07269149	0.072691	✓ matches
Standard error	1.897367	1.89737	✓ matches

One-sample proportion z-test (58 of 100 vs 0.5) · reference: `scipy.stats.norm.sf`
Quantity	Reference value	Our result	Status
Test statistic z	1.6	1.6	✓ matches
p-value	0.1095986	0.1096	✓ matches

One-sample z-test, right-tailed · reference: `scipy.stats.norm.sf`
Quantity	Reference value	Our result	Status
Test statistic z	1.844662	1.8447	✓ matches
p-value	0.03254336	0.032543	✓ matches

Level of Significance Calculator

Critical value and power (α = 0.05, d = 0.5, n = 32) · reference: `scipy.stats.norm.ppf / cdf`
Quantity	Reference value	Our result	Status
Critical values (±)	1.959964	± 1.96	✓ matches
Power (1 − β)	80.74304	80.74 %	✓ matches
Type II error probability β	19.25696	19.26 %	✓ matches

Critical t value (α = 0.01, df = 12, right tail) · reference: `scipy.stats.t.ppf`
Quantity	Reference value	Our result	Status
Critical value	2.680998	2.681	✓ matches

Probability and Chance Calculator

Chance of an event with p = 1/6 · reference: `scipy.stats.binom.pmf, 1 − (1 − p)ⁿ`
Quantity	Reference value	Our result	Status
Probability of the event (single trial)	0.1666667	0.16666667	✓ matches
At least once in 10 trials	0.8384944	0.83849442	✓ matches
Exactly 2 times in 10 trials	0.29071	0.29071005	✓ matches

Raosoft Sample Size Calculator

Raosoft sample size (±5%, 95%, N = 1500) · closed form: `n = N·x / ((N − 1)E² + x), x = z²·r(100 − r)`
Quantity	Reference value	Our result	Status
Recommended sample size	306	306	✓ matches
Without population correction	385	385	✓ matches

Statistical Significance Calculator

A/B test: 250/5000 vs 300/5000 conversions · reference: `scipy.stats.chi2_contingency (z² = χ² for a 2 × 2 table)`
Quantity	Reference value	Our result	Status
p-value	0.02829497	0.028295	✓ matches
z statistic	2.193172	2.193	✓ matches
Relative uplift of B over A	20	20 %	✓ matches

Sample Size Calculator

Two-proportion sample size (30% vs 45%, 80% power) · closed form: `n = [z(α/2)√(2p̄q̄) + z(β)√(p₁q₁ + p₂q₂)]² / (p₁ − p₂)²`
Quantity	Reference value	Our result	Status
Sample size per group (two proportions)	163	163	✓ matches

Percentage Error Calculator

Percentage error · closed form: `(approximate − exact) ÷ exact × 100`
Quantity	Reference value	Our result	Status
Percentage error	-2.140673	-2.14067 %	✓ matches
Absolute percentage error	2.140673	2.14067 %	✓ matches
Absolute error	0.21	0.21	✓ matches

Standard Deviation Calculator for Grouped Data

Grouped data SD (class midpoints) · closed form: `numpy with class midpoints`
Quantity	Reference value	Our result	Status
Mean of grouped data	25.88889	25.8889	✓ matches
Sample standard deviation (s)	11.24565	11.2457	✓ matches
Total frequency (N)	45	45	✓ matches
Σ f(x − x̄)²	5564.444	5564.44	✓ matches

Average Deviation Calculator

Average deviation · reference: `numpy`
Quantity	Reference value	Our result	Status
Average deviation about the mean	3.75	3.75	✓ matches
Average deviation about the median	3.5	3.5	✓ matches

Descriptive Statistics Calculator

Descriptive statistics · reference: `numpy`
Quantity	Reference value	Our result	Status
Count (n)	15	15	✓ matches
Mean	16.33333	16.3333	✓ matches
Median	15	15	✓ matches
Sum	245	245	✓ matches
Range	38	38	✓ matches
Sample variance (s²)	111.2381	111.238	✓ matches

Coefficient of Variation Calculator

Coefficient of variation · reference: `numpy.std(ddof=1) / numpy.mean × 100`
Quantity	Reference value	Our result	Status
Coefficient of variation (CV)	9.819304	9.8193 %	✓ matches
Standard deviation	2.160247	2.16025	✓ matches

Histogram Calculator and Maker

Histogram default bin count (Sturges) · closed form: `numpy.histogram_bin_edges(bins='sturges') uses ⌈log₂ n + 1⌉`
Quantity	Reference value	Our result	Status
Number of bins	5	5	✓ matches

Curve Fitting Calculator

Polynomial fit, degree 1 · reference: `numpy.polyfit(x, y, 1)`
Quantity	Reference value	Our result	Status
R²	0.9463986	0.9464	✓ matches
RMSE	7.506343	7.5063	✓ matches
SSE	563.4519	563.452	✓ matches

Polynomial fit, degree 2 · reference: `numpy.polyfit(x, y, 2)`
Quantity	Reference value	Our result	Status
R²	0.999914	0.99991	✓ matches
RMSE	0.3007541	0.30075	✓ matches
SSE	0.9045303	0.90453	✓ matches

Polynomial fit, degree 3 · reference: `numpy.polyfit(x, y, 3)`
Quantity	Reference value	Our result	Status
R²	0.9999443	0.99994	✓ matches
RMSE	0.242019	0.24202	✓ matches
SSE	0.5857319	0.585732	✓ matches

Exponential fit (log-linear least squares) · reference: `numpy.polyfit(x, ln y, 1), error measured on the original y scale`
Quantity	Reference value	Our result	Status
R²	0.9997947	0.99979	✓ matches
SSE	1.454242	1.45424	✓ matches

Nonlinear Regression Calculator

Logistic curve fit · reference: `scipy.optimize.curve_fit (Levenberg-Marquardt)`
Quantity	Reference value	Our result	Status
Parameter L	76.00643	76.00643	✓ matches
Parameter k	0.7350666	0.7350666	✓ matches
Parameter x₀	5.015195	5.015195	✓ matches
SSE	0.3020473	0.302047	✓ matches

Michaelis-Menten fit · reference: `scipy.optimize.curve_fit (Levenberg-Marquardt)`
Quantity	Reference value	Our result	Status
Parameter Vmax	12.71293	12.71293	✓ matches
Parameter K	2.724184	2.724184	✓ matches
SSE	0.04800794	0.0480079	✓ matches

Decay to a baseline fit · reference: `scipy.optimize.curve_fit (Levenberg-Marquardt)`
Quantity	Reference value	Our result	Status
Parameter a	9.428508	9.428508	✓ matches
Parameter b	0.4687657	0.4687657	✓ matches
Parameter c	2.837218	2.837218	✓ matches

Normal Probability Calculator

Between 85 and 130, N(100, 15) · reference: `scipy.stats.norm.cdf`
Quantity	Reference value	Our result	Status
P(85 < X < 130)	0.8185946	0.81859461	✓ matches
z-score of 85	-1	-1	✓ matches

Right tail above 120 · reference: `scipy.stats.norm.sf`
Quantity	Reference value	Our result	Status
P(X > 120)	0.09121122	0.09121122	✓ matches

Inverse Z Score Calculator

95% middle area · reference: `scipy.stats.norm.ppf(0.975)`
Quantity	Reference value	Our result	Status
z cut-offs enclosing the middle 95 %	1.959964	±1.959964	✓ matches

90th percentile, N(100, 15) · reference: `scipy.stats.norm.ppf`
Quantity	Reference value	Our result	Status
z with 90 % of the area to the left	1.281552	1.2815516	✓ matches
Equivalent x value(s)	119.2233	119.22327	✓ matches

Exponential Probability Calculator

Mean 10, P(X <= 5) · reference: `scipy.stats.expon`
Quantity	Reference value	Our result	Status
P(X ≤ 5)	0.3934693	0.39346934	✓ matches
Median ln(2)/λ	6.931472	6.9314718	✓ matches
Variance 1/λ²	100	100	✓ matches

Rate 0.2, P(X > 3) · reference: `scipy.stats.expon.sf`
Quantity	Reference value	Our result	Status
P(X > 3)	0.5488116	0.54881164	✓ matches

Normal Approximation to the Binomial Calculator

P(X <= 45), n=100, p=0.4 · reference: `scipy.stats.norm.cdf, scipy.stats.binom.cdf`
Quantity	Reference value	Our result	Status
P(X ≤ 45), normal approximation	0.8692139	0.86921388	✓ matches
P(X ≤ 45), exact binomial	0.8689095	0.86890955	✓ matches

Expected Value and Variance Calculator

Discrete distribution · closed form: `NumPy weighted sums`
Quantity	Reference value	Our result	Status
Expected value E[X]	2	2	✓ matches
Variance Var(X)	1.2	1.2	✓ matches

Probability of A Given B Calculator

P(A)=0.3, P(B)=0.4, P(A and B)=0.15 · closed form: `Definition of conditional probability`
Quantity	Reference value	Our result	Status
P(A \| B)	0.375	0.375	✓ matches
P(B \| A)	0.5	0.5	✓ matches
P(A or B)	0.55	0.55	✓ matches

Median and Quartile Calculator

Quartiles, NumPy method=linear · reference: `numpy.percentile(method='linear')`
Quantity	Reference value	Our result	Status
First quartile Q1	37.5	37.5	✓ matches
Median Q2	41	41	✓ matches
Third quartile Q3	45	45	✓ matches
Interquartile range IQR	7.5	7.5	✓ matches

Quartiles, NumPy method=weibull · reference: `numpy.percentile(method='weibull')`
Quantity	Reference value	Our result	Status
First quartile Q1	36	36	✓ matches
Median Q2	41	41	✓ matches
Third quartile Q3	47	47	✓ matches
Interquartile range IQR	11	11	✓ matches

Sxx Calculator (Sums of Squares)

Sums of squares and regression · reference: `scipy.stats.linregress`
Quantity	Reference value	Our result	Status
Sxx = Σ(x − x̄)²	42	42	✓ matches
Syy = Σ(y − ȳ)²	167.795	167.795	✓ matches
Sxy = Σ(x − x̄)(y − ȳ)	83.9	83.9	✓ matches
Slope b₁ = Sxy/Sxx	1.997619	1.997619	✓ matches
Correlation r = Sxy/√(Sxx·Syy)	0.9994195	0.99941947	✓ matches

Poisson Distribution Calculator

P(X = 6), lambda = 4 · reference: `scipy.stats.poisson`
Quantity	Reference value	Our result	Status
P(X = 6)	0.1041956	0.10419563	✓ matches
P(X ≤ 6)	0.889326	0.88932602	✓ matches
P(X ≥ 6)	0.2148696	0.21486961	✓ matches

Range 3 to 8, lambda = 4 · reference: `scipy.stats.poisson.cdf`
Quantity	Reference value	Our result	Status
P(3 ≤ X ≤ 8)	0.7405333	0.74053326	✓ matches

Uniform Distribution Calculator

Uniform(0, 20) · reference: `scipy.stats.uniform`
Quantity	Reference value	Our result	Status
P(5 ≤ X ≤ 12)	0.35	0.35	✓ matches
Variance (b − a)²/12	33.33333	33.333333	✓ matches

Hypothesis Test P-Value Calculator

t statistic 2.31, two-tailed · reference: `scipy.stats (t)`
Quantity	Reference value	Our result	Status
p-value	0.02980124	0.0298012	✓ matches

t statistic -1.8, left-tailed · reference: `scipy.stats (t)`
Quantity	Reference value	Our result	Status
p-value	0.04851634	0.0485163	✓ matches

z statistic 1.96, two-tailed · reference: `scipy.stats (z)`
Quantity	Reference value	Our result	Status
p-value	0.04999579	0.0499958	✓ matches

chi statistic 18.5, right-tailed · reference: `scipy.stats (chi)`
Quantity	Reference value	Our result	Status
p-value	0.02979634	0.0297963	✓ matches

f statistic 3.5, right-tailed · reference: `scipy.stats (f)`
Quantity	Reference value	Our result	Status
p-value	0.0344931	0.0344931	✓ matches

T Table Calculator (T Distribution)

Critical t, df = 15, 95% · reference: `scipy.stats.t.ppf(0.975, 15)`
Quantity	Reference value	Our result	Status
Two-tailed critical value t*	2.13145	±2.13145	✓ matches

t to p, t = 2.1, df = 15 · reference: `scipy.stats.t.sf`
Quantity	Reference value	Our result	Status
Two-tailed p-value	0.05305526	0.053055256	✓ matches
Right-tailed p-value P(T ≥ t)	0.02652763	0.026527628	✓ matches

F Table Calculator (F Distribution)

Critical F(3, 20) at 0.05 · reference: `scipy.stats.f.ppf(0.95, 3, 20)`
Quantity	Reference value	Our result	Status
Critical value F(3, 20) at α = 0.05	3.098391	3.09839	✓ matches

p-value F = 3.5, df (3, 20) · reference: `scipy.stats.f.sf`
Quantity	Reference value	Our result	Status
p-value (right tail) P(F ≥ f)	0.0344931	0.034493104	✓ matches

Independent Event Probability Calculator

Four independent events · closed form: `Multiplication rule`
Quantity	Reference value	Our result	Status
P(all events occur)	0.018	0.018	✓ matches
P(none occur)	0.112	0.112	✓ matches
P(at least one occurs)	0.888	0.888	✓ matches

Covariance Matrix Calculator

Three-variable data · reference: `numpy.cov / numpy.corrcoef`
Quantity	Reference value	Our result	Status
Covariance Height, Weight	105.5	105.5	✓ matches
Correlation Height, Weight	0.9977912	0.99779116	✓ matches
Variance of Age	29.5	29.5	✓ matches

Multiple Regression Analysis Calculator

House price, three predictors · reference: `numpy.linalg.lstsq, scipy.stats.f.sf`
Quantity	Reference value	Our result	Status
R²	0.9874549	0.987455	✓ matches
Adjusted R²	0.9811824	0.981182	✓ matches
F statistic	157.4253	157.425	✓ matches
p-value for the overall model	0.000004298466	0.00000429847	✓ matches
SSE (residual sum of squares)	408.9753	408.975	✓ matches

Mean and SD Calculator

Mean, sample SD and population SD · reference: `numpy.mean / numpy.std (ddof = 1 and 0)`
Quantity	Reference value	Our result	Status
Mean x̄	15	15	✓ matches
Sample SD s	2.581989	2.5819889	✓ matches
Population SD σ (n)	2.44949	2.4494897	✓ matches
Sample variance s²	6.666667	6.6666667	✓ matches

Sample Mean and Sample Standard Deviation Calculator

Sample statistics and 95% CI · reference: `numpy.std(ddof=1), scipy.stats.sem, scipy.stats.t.interval`
Quantity	Reference value	Our result	Status
Sample mean x̄	24.8375	24.8375	✓ matches
Sample standard deviation s	1.527778	1.5277785	✓ matches
Standard error of the mean s/√n	0.5401513	0.54015127	✓ matches
95 % confidence interval for the population mean	23.56025 to 26.11475	23.56025 to 26.11475	✓ matches

Mean and MAD Calculator

Mean absolute and median absolute deviation · reference: `numpy, scipy.stats.median_abs_deviation`
Quantity	Reference value	Our result	Status
Mean absolute deviation (about the mean)	2	2	✓ matches
Median absolute deviation (about the median)	2	2	✓ matches
Population standard deviation σ	2.291288	2.2912878	✓ matches

2 SEM Calculator

Mean 50, SD 8, n 25 · closed form: `SEM = s/sqrt(n)`
Quantity	Reference value	Our result	Status
Standard error of the mean (SEM)	1.6	1.6	✓ matches
2 × SEM	3.2	3.2	✓ matches
Mean ± 2 SEM	46.8 to 53.2	46.8 to 53.2	✓ matches

Raw data · reference: `scipy.stats.sem`
Quantity	Reference value	Our result	Status
Standard error of the mean (SEM)	0.8164966	0.81649658	✓ matches

Standard Deviation Formula Calculator

Sample formula · reference: `numpy.std(ddof=1)`
Quantity	Reference value	Our result	Status
Standard deviation s	2.94392	2.9439203	✓ matches
Variance s²	8.666667	8.6666667	✓ matches
Shortcut formula gives variance	8.666667	8.6666667	✓ matches

Population formula · reference: `numpy.std(ddof=0)`
Quantity	Reference value	Our result	Status
Standard deviation σ	2.687419	2.6874192	✓ matches
Variance σ²	7.222222	7.2222222	✓ matches

Find Probability Calculator

Normal P(X <= 115) · reference: `scipy.stats.norm.cdf`
Quantity	Reference value	Our result	Status
P(X ≤ 115)	0.8413447	0.84134475	✓ matches

Binomial P(X < 6) · reference: `scipy.stats.binom.cdf`
Quantity	Reference value	Our result	Status
P(X < 6)	0.4163708	0.41637083	✓ matches

Poisson P(X >= 7) · reference: `scipy.stats.poisson.sf`
Quantity	Reference value	Our result	Status
P(X ≥ 7)	0.110674	0.11067398	✓ matches

Exponential P(X > 3) · reference: `scipy.stats.expon.sf`
Quantity	Reference value	Our result	Status
P(X > 3)	0.2231302	0.22313016	✓ matches

Uniform range · reference: `scipy.stats.uniform.cdf`
Quantity	Reference value	Our result	Status
P(2 ≤ X ≤ 7)	0.5	0.5	✓ matches

Student t P(X <= 1.5) · reference: `scipy.stats.t.cdf`
Quantity	Reference value	Our result	Status
P(X ≤ 1.5)	0.9177463	0.91774634	✓ matches

ANOVA Table Calculator

Three groups from summary statistics · closed form: `scipy.stats.f; sums of squares from group summaries`
Quantity	Reference value	Our result	Status
F statistic	6.096236	6.09624	✓ matches
p-value	0.005583279	0.00558328	✓ matches
Critical F at α = 0.05	3.284918	3.28492	✓ matches
Effect size η² = SSB/SST	0.2697899	0.26979	✓ matches

Hypothesis Test Calculator

Welch two-sample t from summary · reference: `scipy.stats.ttest_ind_from_stats(equal_var=False)`
Quantity	Reference value	Our result	Status
t statistic	1.992334	1.99233	✓ matches
p-value	0.05067329	0.0506733	✓ matches

One-sample t, right-tailed · reference: `scipy.stats.t.sf`
Quantity	Reference value	Our result	Status
t statistic	1.622882	1.62288	✓ matches
p-value	0.05771859	0.0577186	✓ matches

Two-proportion z-test · reference: `scipy.stats.norm.sf`
Quantity	Reference value	Our result	Status
z statistic	2.70995	2.70995	✓ matches
p-value	0.006729336	0.00672934	✓ matches

Chi Test Calculator

Goodness of fit, equal expected · reference: `scipy.stats.chisquare`
Quantity	Reference value	Our result	Status
Chi-square test statistic χ²	4.32	4.32	✓ matches
p-value	0.2289189	0.228919	✓ matches

Independence, 2 x 3 table · reference: `scipy.stats.chi2_contingency(correction=False)`
Quantity	Reference value	Our result	Status
Chi-square test statistic χ²	8.263794	8.2637944	✓ matches
p-value	0.0160524	0.0160524	✓ matches
Degrees of freedom	2	2	✓ matches

Statistical Sampling Calculator

Margin of error, N = 500, n = 25 · closed form: `Margin of error with finite population correction`
Quantity	Reference value	Our result	Status
Sampling fraction n/N	5	5 %	✓ matches
Margin of error for a proportion (95 %, p = 0.5)	19.1225	±19.12 percentage points	✓ matches
Finite population correction	0.9756556	0.975656	✓ matches

Combination Sum Calculator

Sum of C(10, k) for k = 0..3 · closed form: `math.comb`
Quantity	Reference value	Our result	Status
Σ C(10, k) for k = 0 to 3	176	176	✓ matches
Share of all subsets	17.1875	17.1875 %	✓ matches

Cell Doubling Time Calculator

100,000 to 850,000 cells in 72 h · closed form: `td = t / log2(N/N0)`
Quantity	Reference value	Our result	Status
Doubling time td	23.32012	23.3201 hours	✓ matches
Number of generations (doublings)	3.087463	3.08746	✓ matches
Specific growth rate μ = ln(N/N₀)/t	0.02972314	0.0297231 per hour	✓ matches

Mean and Stdev Calculator

Ten values · reference: `numpy.std / numpy.var`
Quantity	Reference value	Our result	Status
Mean (AVERAGE)	15	15	✓ matches
Sample stdev (STDEV.S)	2.581989	2.5819889	✓ matches
Population stdev (STDEV.P)	2.44949	2.4494897	✓ matches
Sample variance (VAR.S)	6.666667	6.6666667	✓ matches
Population variance (VAR.P)	6	6	✓ matches
Standard error stdev/√n	0.8164966	0.81649658	✓ matches

Mean Mode Calculator

Bimodal data · reference: `numpy.mean / hand count of modes`
Quantity	Reference value	Our result	Status
Mean	5.8	5.8	✓ matches
Median	6	6	✓ matches
Modes	5	5, 7	✓ matches

Mean Average Deviation Calculator

Two data sets · reference: `numpy`
Quantity	Reference value	Our result	Status
Mean of A	14.14286	14.142857	✓ matches
Mean average deviation of A	1.877551	1.877551	✓ matches
Mean of B	24.28571	24.285714	✓ matches
Mean average deviation of B	2.897959	2.8979592	✓ matches
Difference of means (A − B)	-10.14286	-10.142857	✓ matches
Difference in units of average MAD	4.247863	4.2479	✓ matches

Relative Average Deviation Calculator

Titration replicates · reference: `numpy`
Quantity	Reference value	Our result	Status
Relative average deviation (%)	0.08602836	0.0860284 %	✓ matches
Relative average deviation (parts per thousand)	0.8602836	0.860284 ppt	✓ matches
Average deviation (about the mean)	0.0216	0.0216	✓ matches
Relative standard deviation (CV)	0.1105181	0.110518 %	✓ matches

Area Model Calculator

Whole numbers · reference: `exact arithmetic`
Quantity	Reference value	Our result	Status
23 × 14	322	322	✓ matches

Decimals · reference: `exact arithmetic`
Quantity	Reference value	Our result	Status
4.5 × 2.3	10.35	10.35	✓ matches

Binomials · reference: `hand expansion of (2x+3)(x+5)`
Quantity	Reference value	Our result	Status
x² term (a·c)	2	2	✓ matches
x term (a·d + b·c)	13	13	✓ matches
Constant term (b·d)	15	15	✓ matches

Probability Z Score Calculator

z = 1.96 · reference: `scipy.stats.norm`
Quantity	Reference value	Our result	Status
P(Z < z) left tail	0.9750021	0.9750021	✓ matches
P(Z > z) right tail	0.0249979	0.024997895	✓ matches
Two-tailed probability	0.04999579	0.04999579	✓ matches
P(−\|z\| < Z < \|z\|)	0.9500042	0.95000421	✓ matches

Raw score 115, mean 100, SD 15 · reference: `scipy.stats.norm`
Quantity	Reference value	Our result	Status
z-score	1	1	✓ matches
P(Z < z) left tail	0.8413447	0.84134475	✓ matches

Curve Fit Calculator

Growth data · reference: `numpy.polyfit, best R² over six models`
Quantity	Reference value	Our result	Status
R² of that model	0.9997449	0.999745	✓ matches

Test Statistic Calculator

One-sample t · reference: `scipy.stats.t`
Quantity	Reference value	Our result	Status
Test statistic t	2.396286	2.3962862	✓ matches
p-value (two-tailed)	0.02323394	0.0232339	✓ matches
Cohen's d	0.4375	0.4375	✓ matches

Welch t · reference: `scipy.stats.ttest_ind_from_stats`
Quantity	Reference value	Our result	Status
Test statistic t	2.007211	2.0072111	✓ matches
p-value (two-tailed)	0.04973326	0.0497333	✓ matches

Chi-square GOF · reference: `scipy.stats.chisquare`
Quantity	Reference value	Our result	Status
Test statistic χ²	12.32	12.32	✓ matches
p-value (right tail)	0.00636363	0.00636363	✓ matches

One-proportion z · reference: `scipy.stats.norm`
Quantity	Reference value	Our result	Status
Test statistic z	1.6	1.6	✓ matches
p-value (two-tailed)	0.1095986	0.109599	✓ matches

Range of a Function Calculator

Quadratic x^2 - 4x + 3 · reference: `vertex formula`
Quantity	Reference value	Our result	Status
Minimum value	-1	-1	✓ matches

Quadratic on [3, 6] · reference: `endpoint evaluation`
Quantity	Reference value	Our result	Status
Minimum value	0	0	✓ matches
Maximum value	15	15	✓ matches

Binomial Distribution Calculator With Steps

P(2 <= X <= 5) · reference: `scipy.stats.binom`
Quantity	Reference value	Our result	Status
P(2 ≤ X ≤ 5)	0.7971262	0.79712621	✓ matches
Mean n·p	3.6	3.6	✓ matches
Variance n·p·(1 − p)	2.52	2.52	✓ matches
Standard deviation	1.587451	1.5874508	✓ matches

P(X >= 7) · reference: `scipy.stats.binom.sf`
Quantity	Reference value	Our result	Status
P(X ≥ 7)	0.03860084	0.038600843	✓ matches

Average and Standard Deviation Calculator

With an outlier · reference: `numpy`
Quantity	Reference value	Our result	Status
Average (mean)	26.88889	26.888889	✓ matches
Median	25	25	✓ matches
Midrange	33	33	✓ matches
Standard deviation (sample)	11.22992	11.22992	✓ matches
Standard deviation (population)	10.58767	10.58767	✓ matches

Hypothesis Calculator

Two-tailed power · reference: `scipy.stats.norm`
Quantity	Reference value	Our result	Status
Critical z	1.959964	±1.95996	✓ matches
Power if the true mean is 105	0.5160053	0.516005	✓ matches
Type II error β	0.4839947	0.483995	✓ matches
n needed for power 0.8	71	71	✓ matches

SD Formula Calculator

Sample SD from sums · reference: `hand calculation`
Quantity	Reference value	Our result	Status
Sample standard deviation s	15.16575	15.165751	✓ matches
Sample variance s²	230	230	✓ matches
Sum of squares SS	1150	1150	✓ matches

Population SD from frequency table · reference: `numpy.std on expanded data`
Quantity	Reference value	Our result	Status
Population standard deviation σ	1.943067	1.9430672	✓ matches
Mean	4.714286	4.7142857	✓ matches

Median Score Calculator

Ten test scores · reference: `numpy / scipy.stats.percentileofscore`
Quantity	Reference value	Our result	Status
Median score	80	80	✓ matches
Mean score	80	80	✓ matches
Percentile rank of 85	65	65 %	✓ matches

Calculator With Standard Deviation

One-variable stats · reference: `numpy`
Quantity	Reference value	Our result	Status
Sample standard deviation Sx	2.213594	2.2135944	✓ matches
Population standard deviation σx	2.1	2.1	✓ matches
Mean x̄	6.3	6.3	✓ matches
Σx²	441	441	✓ matches

Mean, Median and Standard Deviation Calculator

Data with an outlier · reference: `numpy`
Quantity	Reference value	Our result	Status
Mean	21.6	21.6	✓ matches
Median	17	17	✓ matches
Standard deviation (sample)	15.22571	15.225709	✓ matches
Standard deviation (population)	14.44438	14.444376	✓ matches
Mean change if the largest value is dropped	-4.711111	-4.7111111	✓ matches
Median change if the largest value is dropped	-1	-1	✓ matches

Standard Deviation From Mean and Sample Size Calculator

From SEM · reference: `SD = SEM x sqrt(n)`
Quantity	Reference value	Our result	Status
Standard deviation	9	9	✓ matches

From a 95% CI · reference: `scipy.stats.t.ppf`
Quantity	Reference value	Our result	Status
Standard deviation	8.963619	8.9636188	✓ matches
Mean	50	50	✓ matches

From the CV · reference: `SD = mean x CV`
Quantity	Reference value	Our result	Status
Standard deviation	6	6	✓ matches
Coefficient of variation	12	12 %	✓ matches

From the sum of squares · reference: `SS = sum(x^2) - n mean^2`
Quantity	Reference value	Our result	Status
Standard deviation	6.123724	6.1237244	✓ matches

Rearrange Formula Calculator

Sample size from margin of error · reference: `algebra`
Quantity	Reference value	Our result	Status
n =	138.2976	138.2976	✓ matches

x from a z-score · reference: `algebra`
Quantity	Reference value	Our result	Status
x =	122.5	122.5	✓ matches

n from a t statistic · reference: `algebra`
Quantity	Reference value	Our result	Status
n =	56.25	56.25	✓ matches

Statistics Confidence Interval Calculator

Mean t interval · reference: `scipy.stats.t.interval`
Quantity	Reference value	Our result	Status
95 % confidence interval	49.37541	49.3754 to 55.4246	✓ matches
Margin of error	3.02459	3.02459	✓ matches
t critical value	2.04523	2.04523	✓ matches

Wilson interval · reference: `scipy.stats.binomtest.proportion_ci(wilson)`
Quantity	Reference value	Our result	Status
95 % confidence interval	0.4820649	0.482065 to 0.672016	✓ matches
Sample proportion p̂	0.58	0.58	✓ matches

Indicated Probability Calculator

Red or blue · reference: `exact fraction`
Quantity	Reference value	Our result	Status
P(indicated event)	0.5714286	0.57142857	✓ matches
Probability of the opposite	0.4285714	0.42857143	✓ matches
Favourable outcomes	8	8	✓ matches
Total outcomes	14	14	✓ matches

Simple Regression Calculator

Ten points · reference: `scipy.stats.linregress`
Quantity	Reference value	Our result	Status
Slope b₁	2.007273	2.0072727	✓ matches
Intercept b₀	0.06	0.06	✓ matches
Correlation r	0.9989253	0.998925	✓ matches
R²	0.9978517	0.997852	✓ matches
Standard error of the estimate	0.2990895	0.29909	✓ matches
Slope standard error	0.03292867	0.0329287	✓ matches
Slope t statistic	60.9582	60.9582	✓ matches
95% CI for the slope	1.931339	1.93134 to 2.08321	✓ matches
Predicted y at x = 11	22.14	22.14	✓ matches
95% prediction interval	21.30473	21.3047 to 22.9753	✓ matches

Probability Calculator for 1 Event

Rolling a six · reference: `scipy.stats.binom`
Quantity	Reference value	Our result	Status
P(event) on one trial	0.1666667	0.16666667	✓ matches
P(at least once in 10 trials)	0.8384944	0.83849442	✓ matches
P(exactly 2 times in 10 trials)	0.29071	0.29071005	✓ matches
Expected number of times	1.666667	1.6666667	✓ matches
Trials for a 50 % chance of at least one	4	4	✓ matches
Trials for a 95 % chance of at least one	17	17	✓ matches

Range Symbol on Calculator

Eight values · reference: `numpy`
Quantity	Reference value	Our result	Status
Range R = max − min	27	27	✓ matches
Maximum	45	45	✓ matches
Minimum	18	18	✓ matches
Midrange	31.5	31.5	✓ matches
Interquartile range	13	13	✓ matches
Range ÷ sample SD	3.032601	3.0326	✓ matches

Standard Deviation Index Calculator

One result · reference: `scipy.stats.norm`
Quantity	Reference value	Our result	Status
Standard deviation index (SDI)	1.68	1.68	✓ matches
Difference from the group mean	4.2	4.2	✓ matches
Bias as % of the group mean	4.2	4.2 %	✓ matches
Share of a normal group below this result	95.35213	95.352 %	✓ matches

Histogram Mean Calculator

Grouped data · reference: `class-midpoint formulas`
Quantity	Reference value	Our result	Status
Estimated mean	25.6	25.6	✓ matches
Estimated median	25.55556	25.555556	✓ matches
Estimated mode	25.38462	25.384615	✓ matches
Estimated standard deviation (sample)	10.95631	10.956314	✓ matches
Estimated standard deviation (population)	10.8462	10.846197	✓ matches

Coefficient of Skewness Calculator

Right-skewed data · reference: `scipy.stats.skew`
Quantity	Reference value	Our result	Status
Pearson second coefficient 3(mean − median)/s	0.9121035	0.912103	✓ matches
Pearson first coefficient (mean − mode)/s	0.4560517	0.456052	✓ matches
Bowley coefficient (quartile)	0.3333333	0.333333	✓ matches
Moment coefficient g1	1.537067	1.53707	✓ matches
Adjusted coefficient G1 (sample)	1.765955	1.76595	✓ matches

Find the Margin of Error Calculator

Poll proportion · reference: `scipy.stats.norm.ppf`
Quantity	Reference value	Our result	Status
Margin of error	0.03096495	0.03096495	✓ matches
Standard error	0.01579873	0.015798734	✓ matches
Critical value	1.959964	1.95996	✓ matches

Mean with t · reference: `scipy.stats.t.ppf`
Quantity	Reference value	Our result	Status
Margin of error	0.7446562	0.74465623	✓ matches
Critical value	1.962341	1.96234	✓ matches

From an interval · reference: `(U - L)/2`
Quantity	Reference value	Our result	Status
Margin of error	3.1	3.1	✓ matches
Point estimate (centre)	52	52	✓ matches
Interval width	6.2	6.2	✓ matches

Polynomial Curve Fitting Calculator

Quadratic · reference: `numpy.polyfit`
Quantity	Reference value	Our result	Status
R²	0.9999094	0.999909	✓ matches
Adjusted R²	0.9998792	0.999879	✓ matches
RMSE	0.3200349	0.320035	✓ matches
Prediction at x = 9	128.5524	128.552	✓ matches

Area Model Multiplication Calculator

Two-digit · reference: `exact arithmetic`
Quantity	Reference value	Our result	Status
36 × 24	864	864	✓ matches
Number of partial products	4	4	✓ matches

Decimals · reference: `exact arithmetic`
Quantity	Reference value	Our result	Status
4.5 × 2.3	10.35	10.35	✓ matches

Variance of a Probability Distribution Calculator

Distribution binom · reference: `scipy.stats distribution moments`
Quantity	Reference value	Our result	Status
Variance Var(X)	4.2	4.2	✓ matches
Standard deviation	2.04939	2.0493902	✓ matches
Mean E(X)	6	6	✓ matches

Distribution pois · reference: `scipy.stats distribution moments`
Quantity	Reference value	Our result	Status
Variance Var(X)	3.5	3.5	✓ matches
Standard deviation	1.870829	1.8708287	✓ matches
Mean E(X)	3.5	3.5	✓ matches

Distribution geom · reference: `scipy.stats distribution moments`
Quantity	Reference value	Our result	Status
Variance Var(X)	12	12	✓ matches
Standard deviation	3.464102	3.4641016	✓ matches
Mean E(X)	4	4	✓ matches

Distribution udisc · reference: `scipy.stats distribution moments`
Quantity	Reference value	Our result	Status
Variance Var(X)	2.916667	2.9166667	✓ matches
Standard deviation	1.707825	1.7078251	✓ matches
Mean E(X)	3.5	3.5	✓ matches

Distribution ucont · reference: `scipy.stats distribution moments`
Quantity	Reference value	Our result	Status
Variance Var(X)	3	3	✓ matches
Standard deviation	1.732051	1.7320508	✓ matches
Mean E(X)	5	5	✓ matches

Distribution exp · reference: `scipy.stats distribution moments`
Quantity	Reference value	Our result	Status
Variance Var(X)	4	4	✓ matches
Standard deviation	2	2	✓ matches
Mean E(X)	2	2	✓ matches

Distribution norm · reference: `scipy.stats distribution moments`
Quantity	Reference value	Our result	Status
Variance Var(X)	9	9	✓ matches
Standard deviation	3	3	✓ matches
Mean E(X)	10	10	✓ matches

How to Find the Mean Calculator

List · reference: `arithmetic`
Quantity	Reference value	Our result	Status
Mean	77.8	77.8	✓ matches
Sum of the values	389	389	✓ matches

Target mean · reference: `80 x 6 - 389`
Quantity	Reference value	Our result	Status
Next value needed for a mean of 80	91	91	✓ matches

Frequency table · reference: `numpy.mean on expanded data`
Quantity	Reference value	Our result	Status
Mean	2.76	2.76	✓ matches
Total frequency n	25	25	✓ matches

Variance Symbol Calculator

Sample and population variance · reference: `numpy.var / numpy.std with ddof = 1 and 0`
Quantity	Reference value	Our result	Status
Sample variance s²	4.666667	4.6666667	✓ matches
Population variance σ²	4	4	✓ matches
Sample standard deviation s	2.160247	2.1602469	✓ matches
Population standard deviation σ	2	2	✓ matches

Combination Probability Calculator

Five cards, exactly two hearts · reference: `scipy.stats.hypergeom(M=52, n=13, N=5)`
Quantity	Reference value	Our result	Status
P(exactly 2 special)	0.2742797	0.27427971	✓ matches
P(at most 2 special)	0.9072329	0.90723289	✓ matches
P(at least 2 special)	0.3670468	0.36704682	✓ matches
Expected number of special items	1.25	1.25	✓ matches

Mean SD CV Calculator

Two sets compared · reference: `numpy.mean / numpy.std(ddof=1)`
Quantity	Reference value	Our result	Status
Mean of A	52.71429	52.714286	✓ matches
Standard deviation of A (sample)	3.147183	3.1471832	✓ matches
CV of A	5.970266	5.97027 %	✓ matches
Mean of B	227.5714	227.57143	✓ matches
Standard deviation of B (sample)	21.82397	21.823971	✓ matches
CV of B	9.589943	9.58994 %	✓ matches

Standard Error Calculator

Standard error from raw data · reference: `scipy.stats.sem`
Quantity	Reference value	Our result	Status
Standard error	2.312621	2.3126207	✓ matches
Sample standard deviation	6.541079	6.541079	✓ matches

Standard error from SD and n · reference: `8 / sqrt(50)`
Quantity	Reference value	Our result	Status
Standard error	1.131371	1.1313708	✓ matches

Standard error of a proportion · reference: `sqrt(p(1 − p) / n)`
Quantity	Reference value	Our result	Status
Standard error	0.06928203	0.069282032	✓ matches

Basis for Range of Matrix Calculator

Rank of a 3 × 4 matrix · reference: `numpy.linalg.matrix_rank`
Quantity	Reference value	Our result	Status
Rank = dimension of the column space	2	2	✓ matches
Nullity (columns − rank)	2	2	✓ matches

Exponential Decay Model Calculator

Half-life 12, N0 = 500 · reference: `N0·exp(−kt), k = ln 2 / half-life`
Quantity	Reference value	Our result	Status
Decay constant k	0.05776227	0.057762265	✓ matches
Amount remaining at t = 30	88.38835	88.388348	✓ matches
Time to fall to 50	39.86314	39.863137	✓ matches

Linear Equations With Distribution Calculator

3(2x − 5) + 4 = 2(x + 3) − 1 · reference: `numpy.linalg.solve on 4x = 16`
Quantity	Reference value	Our result	Status
Solution	4	x = 4	✓ matches

Binomial Expansion of (1 + x)ⁿ Calculator

(1 + x)^6 at x = 0.1 · reference: `math.comb and 1.1**6`
Quantity	Reference value	Our result	Status
Sum of coefficients 2ⁿ	64	64	✓ matches
Middle coefficient	20	20	✓ matches
Value at x = 0.1	1.771561	1.771561	✓ matches

Central Limit Theorem Calculator

Exponential population, n = 30 · reference: `scipy.stats.norm.cdf((x − μ) / (σ / √n))`
Quantity	Reference value	Our result	Status
Standard error σ ÷ √n	0.1825742	0.182574	✓ matches
Mean of the sampling distribution	1	1	✓ matches
P(sample mean ≤ 1.2) by the CLT	0.8633392	0.86334	✓ matches

Critical Value of z Calculator

95% confidence, two-tailed · reference: `scipy.stats.norm.ppf`
Quantity	Reference value	Our result	Status
Critical values of z	1.959964	± 1.95996	✓ matches

99% confidence, two-tailed · reference: `scipy.stats.norm.ppf`
Quantity	Reference value	Our result	Status
Critical values of z	2.575829	± 2.57583	✓ matches

90% confidence, right-tailed · reference: `scipy.stats.norm.ppf`
Quantity	Reference value	Our result	Status
Critical value of z	1.281552	1.28155	✓ matches

Left-tailed α = 0.01 · reference: `scipy.stats.norm.ppf(0.01)`
Quantity	Reference value	Our result	Status
Critical value of z	-2.326348	-2.32635	✓ matches

Moment Generating Function Calculator

Binomial(10, 0.3) at t = 0.5 · reference: `closed form (1 − p + p·eᵗ)ⁿ`
Quantity	Reference value	Our result	Status
M(0.5)	5.919495	5.919495	✓ matches
Mean E[X] = M′(0)	3	3	✓ matches
Variance Var(X) = M″(0) − M′(0)²	2.1	2.1	✓ matches

Normal(2, 3) at t = 0.4 · reference: `closed form exp(μt + σ²t²/2)`
Quantity	Reference value	Our result	Status
M(0.4)	4.572225	4.5722252	✓ matches
Variance Var(X) = M″(0) − M′(0)²	9	9	✓ matches

Exponential(rate 2) at t = 0.5 · reference: `closed form λ / (λ − t)`
Quantity	Reference value	Our result	Status
M(0.5)	1.333333	1.3333333	✓ matches
Mean E[X] = M′(0)	0.5	0.5	✓ matches
Variance Var(X) = M″(0) − M′(0)²	0.25	0.25	✓ matches

Poisson(3) at t = 0.5 · reference: `closed form exp(λ(eᵗ − 1))`
Quantity	Reference value	Our result	Status
M(0.5)	7.001776	7.0017759	✓ matches
Mean E[X] = M′(0)	3	3	✓ matches
Variance Var(X) = M″(0) − M′(0)²	3	3	✓ matches

Chi-Square Table Calculator

df = 4, α = 0.05 · reference: `scipy.stats.chi2.ppf`
Quantity	Reference value	Our result	Status
Right-tail critical value	9.487729	9.48773	✓ matches
Left-tail critical value	0.710723	0.710723	✓ matches

df = 10, α = 0.01 · reference: `scipy.stats.chi2.ppf`
Quantity	Reference value	Our result	Status
Right-tail critical value	23.20925	23.2093	✓ matches
Left-tail critical value	2.558212	2.55821	✓ matches

df = 25, α = 0.1 · reference: `scipy.stats.chi2.ppf`
Quantity	Reference value	Our result	Status
Right-tail critical value	34.38159	34.3816	✓ matches
Left-tail critical value	16.47341	16.4734	✓ matches

p-value for a statistic · reference: `scipy.stats.chi2.sf(11.2, 4)`
Quantity	Reference value	Our result	Status
p-value for your χ²	0.0244059	0.024406	✓ matches

Linear Correlation Coefficient Calculator

Weakly correlated pairs · reference: `scipy.stats.pearsonr / spearmanr, Fisher z interval`
Quantity	Reference value	Our result	Status
Pearson correlation r	0.6180168	0.618017	✓ matches
Coefficient of determination r²	0.3819447	0.381945	✓ matches
t statistic	2.079867	2.0799	✓ matches
p-value (two-tailed)	0.07610004	0.0761	✓ matches
95% confidence interval for ρ	-0.07820205 to 0.9090355	-0.078202 to 0.90904	✓ matches
Spearman rank correlation	0.5666667	0.566667	✓ matches

Least Squares Calculator

Six-point least-squares line · reference: `scipy.stats.linregress`
Quantity	Reference value	Our result	Status
Slope b₁	2.02	2.02	✓ matches
Intercept b₀	-0.02	-0.02	✓ matches
Sum of squared errors (SSE)	0.128	0.128	✓ matches
R²	0.9982107	0.998211	✓ matches
Prediction at x = 8	16.14	16.14	✓ matches

Raosoft Sample Size Calculator Citation Helper

N = 1500, ±5%, 95% · reference: `n = N·x / ((N − 1)E² + x), rounded up`
Quantity	Reference value	Our result	Status
Minimum sample size	306	306	✓ matches

N = 20000, ±3%, 99% · reference: `n = N·x / ((N − 1)E² + x), rounded up`
Quantity	Reference value	Our result	Status
Minimum sample size	1437	1437	✓ matches

Binomial Calculator with Steps

Percentile k · reference: `scipy.stats.binom.ppf(0.9, 40, 0.3)`
Quantity	Reference value	Our result	Status
Smallest k with P(X ≤ k) ≥ 0.9	16	16	✓ matches

Most likely outcome · reference: `argmax of scipy.stats.binom.pmf`
Quantity	Reference value	Our result	Status
Most likely number of successes	12	12	✓ matches

Trials needed for at least 3 successes · reference: `scipy.stats.binom.sf search`
Quantity	Reference value	Our result	Status
Smallest n	19	19	✓ matches
P(at least 3 successes) at n = 19	0.9537763	0.953776	✓ matches

Accuracy verification

How the check works

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