This Python notebook describes several statistical tests and concepts, presented in alphabetical order.

# Import modules
import numpy as np
import pandas as pd
import seaborn as sns
import warnings
import random
import matplotlib.pyplot as plt 
warnings.filterwarnings('ignore')

Anderson Darling Normality Test

This test determines if a data set is normally distributed.

from scipy.stats import anderson

normally_distibuted_data = np.random.normal(size=1000)

anderson_result = anderson(normally_distibuted_data)

significance_level, critical_value = anderson_result.significance_level[2], anderson_result.critical_values[2]
if anderson_result.statistic < critical_value:
   print('Data follows normal distribution (p ≤ 0.05).')
else:
   print('Data does not follows normal distribution')
print("The statistic is: "  + str(anderson_result.statistic))

Data follows normal distribution (p ≤ 0.05).
The statistic is: 0.5202687985699868

ANOVA (and Tukey)

Independent one way

The ANOVA is used to determine whether there are any statistically significant differences between the means of three or more independent groups. The Tukey test is used to determine where the differences lie.

from scipy.stats import f_oneway # for ANOVA
from statsmodels.stats.multicomp import pairwise_tukeyhsd # for Tukey test

# Create data
df = pd.DataFrame({'score': [85, 86, 88, 75, 78, 94, 98, 79, 71, 80, # group a
                             91, 92, 93, 90, 97, 94, 82, 88, 95, 96, # group b
                             79, 78, 88, 94, 92, 85, 83, 85, 82, 81], # group c
                   'group': np.repeat(['a', 'b', 'c'], repeats=10)}) 

a = df.score[df.group == 'a'].tolist()
b = df.score[df.group == 'b'].tolist()
c = df.score[df.group == 'c'].tolist()

# Perform ANOVA
statistic, pvalue = f_oneway(a, b, c)
if pvalue >= 0.05:
    print("There is no significant difference between the means.")
else:
    print("There is a significant difference between the means.")
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue) + "\n")

# Perform Tukey's test
tukey = pairwise_tukeyhsd(endog=df['score'], groups=df['group'], alpha=0.05)
print(tukey)

There is a significant difference between the means.
The statistic is: 5.167774552944481, the p-value is: 0.012582197136592605

 Multiple Comparison of Means - Tukey HSD, FWER=0.05 
=====================================================
group1 group2 meandiff p-adj   lower    upper  reject
-----------------------------------------------------
     a      b      8.4 0.0159   1.4259 15.3741   True
     a      c      1.3 0.8894  -5.6741  8.2741  False
     b      c     -7.1 0.0454 -14.0741 -0.1259   True
-----------------------------------------------------

Independent two way

This test compares the mean differences between groups that have been split on two independent variables (factors). This test determines if there is an interaction between the two independent variables on the dependent variable.

from statsmodels.formula.api import ols
from statsmodels.stats.anova import anova_lm

# Create data
df = pd.DataFrame({'water': np.repeat(['daily', 'weekly'], 15),
                   'sun': np.tile(np.repeat(['low', 'med', 'high'], 5), 2),
                   'height': [6, 6, 6, 5, 6, 5, 5, 6, 4, 5,
                              6, 6, 7, 8, 7, 3, 4, 4, 4, 5,
                              4, 4, 4, 4, 4, 5, 6, 6, 7, 8]})
# Perform ANOVA
model = ols('height ~ C(water) + C(sun) + C(water):C(sun)', data=df).fit()
print(anova_lm(model, typ=2))

                    sum_sq    df        F    PR(>F)
C(water)          8.533333   1.0  16.0000  0.000527
C(sun)           24.866667   2.0  23.3125  0.000002
C(water):C(sun)   2.466667   2.0   2.3125  0.120667
Residual         12.800000  24.0      NaN       NaN

Within-subjects one way

This test is used to compare three or more group means where the participants are the same in each group.

from statsmodels.stats.anova import AnovaRM

# Create data
data = {
    'subject': ['S1'] * 6 + ['S2'] * 6 + ['S3'] * 6,
    'time': ['T1', 'T2', 'T3'] * 6,
    'condition': ['C1', 'C1', 'C1', 'C2', 'C2', 'C2'] * 3,
    'performance': [
        0.1, 0.2, 0.3,  # subject 1, condition 1
        0.1, 0.2, 0.4,  # subject 1, condition 2
        0.3, 0.3, 0.4,  # subject 2, condition 1
        0.3, 0.4, 0.5,  # subject 2, condition 2
        0.5, 0.6, 0.7,  # subject 3, condition 1
        0.6, 0.6, 0.8   # subject 3, condition 2
    ]
}
df = pd.DataFrame(data)

# Perform ANOVA
aovrm = AnovaRM(df, 'performance', 'subject', within=['time', 'condition'])
res = aovrm.fit()
print(res.summary())

                   Anova
===========================================
               F Value Num DF Den DF Pr > F
-------------------------------------------
time           44.8000 2.0000 4.0000 0.0018
condition      25.0000 1.0000 2.0000 0.0377
time:condition  1.6000 2.0000 4.0000 0.3086
===========================================

Augmented Dickey-Fuller Test

This test determines if a time series has stationarity (constant mean, variance, and covariance over time).

# Create 2-column dataframe with 'Date' and 'Sales'
data = {'Date': ['12/02/2024', '13/02/2024', '14/02/2024', '15/02/2024', '16/02/2024', '17/02/2024', '18/02/2024', '19/02/2024', '20/02/2024', '21/02/2024', '22/02/2024', '23/02/2024', '24/02/2024', '25/02/2024', '26/02/2024', '27/02/2024', '28/02/2024', '29/02/2024', '01/03/2024', '02/03/2024', '03/03/2024', '04/03/2024'],
              'Sales': [1757, 1844, 1612, 1742, 1144, 1778, 1105, 1388, 1936, 1119, 850, 812, 812, 848, 763, 781, 794, 792, 780, 763, 785, 828]}
df = pd.DataFrame(data)

sns.lineplot(x=df['Date'], y=df['Sales'])
plt.title('Sales Over Time', size=15)
plt.xlabel('Time')
plt.ylabel('Sales')
plt.xticks(rotation=90);

from statsmodels.tsa.stattools import adfuller

statistic, pvalue, lags, obs, crit, t = adfuller(df['Sales'])
if pvalue >= 0.05:
 print('Data does not exhibit stationarity.')
else:
 print('Data does exhibit stationarity.')
print("The statistic is: "  + str(statistic) + ", the p-value is: " + str(pvalue))

Data does not exhibit stationarity.
The statistic is: 1.1559905031499378, the p-value is: 0.9956672577818104

Barnard's Test

This test is used to determine if there is a significant association between two categorical variables in a 2x2 contingency table.

from scipy.stats import barnard_exact

# Vaccine (experiment) group
exp_infected = 9
exp_uninfected = 91

# Placebo (control) group
ctrl_infected = 19   
ctrl_uninfected = 81

result = barnard_exact(
    [[exp_infected, ctrl_infected],
    [exp_uninfected, ctrl_uninfected]]
    )

if result.pvalue >= 0.05:
    print("There is no significant difference between the groups.")
else:
    print("There is a significant difference between the groups.")
print("The statistic is: " + str(round(result.statistic,3)) + ", the p-value is: " + str(round(result.pvalue,3)))

There is a significant difference between the groups.
The statistic is: -2.038, the p-value is: 0.044

Bland-Altman plot

A Bland-Altman plot is a scatter plot where the differences between two measurements are plotted against their averages.

from pingouin import plot_blandaltman

df = pd.DataFrame({'Before Treatment': [91.3, 107.3, 118.2, 132.3, 115.6, 97.4, 113, 85.1, 95.3, 106.6, 80.2, 91.9, 83.7, 117.8, 123.9],
                    'After Treatment': [148.8, 118.2, 112.2, 146.3, 129.2, 79.7, 89.3, 69.1, 135.8, 112.4, 116.8, 97.9, 99.8, 99.4, 106.9]})

ax = plot_blandaltman(df['Before Treatment'], df['After Treatment'])

Bartlett’s Test

This test determines if the variances in data sets equal.

from scipy.stats import bartlett

A = [85, 86, 88, 75, 78, 94, 98, 79, 71, 80]
B = [91, 92, 93, 85, 87, 84, 82, 88, 95, 96]
C = [79, 78, 88, 94, 92, 85, 83, 85, 82, 81]

statistic, pvalue = bartlett(A, B, C)
if pvalue >= 0.05:
    print("There is no significant difference between the variances.")
else:
    print("There is a significant difference between the variances.")
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))

There is no significant difference between the variances.
The statistic is: 4.855399204876762, the p-value is: 0.08823958541402636

Breusch-Pagan Test

The Breusch-Pagan Test is used to determine whether or not heteroscedasticity (unequal variance of errors) is present in a regression model.

from statsmodels.formula.api import ols
from statsmodels.stats.diagnostic import het_breuschpagan

# Create and then plot data
X = np.linspace(-2, 5, 100)
y = 2 * X + np.random.normal(scale=1.0, size=100)
df = pd.DataFrame({'X': X, 'y': y})

# Fit the OLS regression model
model = ols('y ~ X', data=df).fit()

# Plot data
fig, ax = plt.subplots(figsize=(7, 6))
ax.scatter(df['X'], df['y'], alpha=0.7, label='Data')
ax.plot(X, model.params['Intercept'] + model.params['X'] * X, color='orange', label='OLS Fit')
ax.set_title('Regression with End-of-Range Outliers')
ax.legend()
plt.tight_layout()
plt.show()

# Perform the Breusch-Pagan test
test_results = het_breuschpagan(model.resid, model.model.exog)
pvalue = test_results[3]
if pvalue >= 0.05:
    print("Homoscedasticity is present, p-value is: " + str(pvalue))
else:
    print("Heteroscedasticity is present, p-value is: " + str(pvalue))

Homoscedasticity is present, p-value is: 0.6643998662503177

Chi-Square (goodness of fit)

The chi-square goodness of fit test assesses the differences between the observed and expected proportions.

from scipy.stats import chisquare

observed_data = [8, 6, 10, 7, 8, 11, 9] 
expected_data = [9, 8, 11, 8, 10, 7, 6] 

statistic, pvalue = chisquare(observed_data, expected_data)
if pvalue >= 0.05:
    print("There is no significant difference the observed and expected data.")
else:
    print("There is a significant difference the observed and expected data.")
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))

There is no significant difference the observed and expected data.
The statistic is: 5.0127344877344875, the p-value is: 0.542180861413329

Cohen's Kappa

This test determines the level of agreement between two raters/judges who each classify items into categories. To determine the level agreement between more than two rates/judges, use Fleiss' Kappa.

from sklearn.metrics import cohen_kappa_score

rater1 = ["negative", "positive", "negative", "neutral", "positive", "negative", "neutral", "positive"]
rater2 = ["positive", "positive", "negative", "neutral", "neutral", "negative", "neutral", "positive"]

kappa = cohen_kappa_score(rater1, rater2)
print("Cohen's Kappa score:", kappa)

if kappa < 0.1:
    print("No agreement.")
elif kappa < 0.2:
    print("Slight agreement.")
elif kappa < 0.4:
    print("Fair agreement.")
elif kappa < 0.6:
    print("Moderate agreement.")
elif kappa < 0.8:
    print("Substantial agreement.")
elif kappa < 0.99:
    print("Near perfect agreement.")
else:
    print("Perfect agreement.")

Cohen's Kappa score: 0.627906976744186
Substantial agreement.

Cronbach's Alpha

This test measures the internal consistency, or reliability, of a set of survey items (such as Likert scale ratings).

from pingouin import cronbach_alpha

df = pd.DataFrame({
    '1': [1, 2, 2, 5, 5, 3, 2, 4],
    '2': [2, 2, 2, 4, 5, 1, 3, 3],
    '3': [3, 2, 3, 3, 5, 2, 3, 5],
})

statistic = cronbach_alpha(df)[0]

print("The statistic is: " + str(statistic))

if statistic < 0.5:
    print("Unacceptable internal consistency.")
elif statistic < 0.6:
    print("Poor internal consistency.")
elif statistic < 0.7:
    print("Questionable internal consistency.")
elif statistic < 0.8:
    print("Acceptable internal consistency.")
elif statistic < 0.9:
    print("Good internal consistency.")
else:
    print("Excellent internal consistency.")

The statistic is: 0.8546511627906975
Good internal consistency.

D'Agostino's K-squared Test

This test determines if a data set is normally distributed.

from scipy.stats import normaltest

normally_distibuted_data = np.random.normal(size=1000)

statistic, pvalue = normaltest(normally_distibuted_data)

if pvalue >= 0.05:
 print('Data follows normal distribution.')
else:
 print('Data does not follow normal distribution.')
print("The statistic is: "  + str(statistic) + ", the p-value is: " + str(pvalue))

Data follows normal distribution.
The statistic is: 1.9871576652651162, the p-value is: 0.3702492569836643

Fisher's Test

This test is used to determine if there is a significant association between two categorical variables in a 2x2 contingency table.

from scipy.stats import fisher_exact

# Vaccine (experiment) group
exp_infected = 8
exp_uninfected = 92

# Placebo (control) group
ctrl_infected = 19   
ctrl_uninfected = 81

result = fisher_exact(
    [[exp_infected, ctrl_infected],
    [exp_uninfected, ctrl_uninfected]]
    )

if result[1] >= 0.05:
    print("There is no significant difference between the groups.")
else:
    print("There is a significant difference between the groups.")
print("The statistic is: " + str(round(result[0],3)) + ", the p-value is: " + str(round(result[1],3)))

There is a significant difference between the groups.
The statistic is: 0.371, the p-value is: 0.037

Friedman Test

Friedman's test is a nonparametric test that compares three or more paired groups.

from scipy.stats import friedmanchisquare

measure1 = [4, 6, 3, 4, 3, 2, 2, 7, 6, 5]
measure2 = [5, 6, 8, 5, 7, 8, 4, 6, 4, 5]
measure3 = [2, 4, 4, 3, 2, 2, 2, 4, 3, 2]

statistic, pvalue = friedmanchisquare(measure1, measure2, measure3)
if pvalue >= 0.05:
    print("There is no significant difference between the measures.")
else:
    print("There is a significant difference between the measures.")
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue)) 

There is a significant difference between the measures.
The statistic is: 12.666666666666673, the p-value is: 0.0017761035457343726

Grubbs’ Test

Grubb's test is used to detect the presence of one outlier in a data set that is normally distributed.

from scikit_posthocs import outliers_grubbs

data = [5.1, 5.2, 5.5, 4.9, 5.3, 70.0, 6400.3]

data_without_outlier = outliers_grubbs(data, alpha=0.05)

if len(data) == len(data_without_outlier):
    print("There is no outlier")
else:
    non_common = list(set(data) ^ set(data_without_outlier))
    print("This is an outlier: " + str(non_common[0]))

This is an outlier: 6400.3

Jarque-Bera test (of normality)

This test determines if a data set is normally distributed.

from scipy.stats import jarque_bera

normally_distibuted_data  = np.random.normal(loc=0, scale=1, size=80)

statistic, pvalue = jarque_bera(normally_distibuted_data)

if pvalue >= 0.05:
    print("Data is normally distributed.")
else:
    print("Data is not normally distributed.")
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue) + "\n")

Data is normally distributed.
The statistic is: 0.050941822676446, the p-value is: 0.9748507356566426

Kendall's Tau Correlation

Kendall's Tau is a non-parametric measure of relationships between columns of ranked data.

from scipy.stats import kendalltau

data = {
    'candidate': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L'],
    'ranker1': [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12],
    'ranker2': [3, 2, 6, 4, 1, 5, 8, 7, 10, 12, 9, 11 ]
}
df = pd.DataFrame(data)
 

statistic, pvalue = kendalltau(df['ranker1'], df['ranker2'])
if pvalue > 0.05:
    print('There is not a relationship between the two rankers.')
else:
    print('There is a relationship between the two rankers.')
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))

There is a relationship between the two rankers.
The statistic is: 0.6666666666666666, the p-value is: 0.0018032758136924804

Kolmogorov-Smirnov Tests

Test of normality

This test determines if data is normally distributed.

from scipy.stats import kstest

data = random.sample(range(0, 99), 80) # assign 80 random (not normally distributed) to list

statistic, pvalue = kstest(data, 'norm')
if pvalue > 0.05:
    print("Data is likely normally distributed.")
else:
    print("Data unlikely to be normally distributed.")
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))

Data unlikely to be normally distributed.
The statistic is: 0.9861501019683699, the p-value is: 4.137672119127385e-149

Two sample test

This test determines whether two samples come from the same distribution.

from scipy.stats import ks_2samp

data1 = np.random.normal(100, 10, 500)  # mean, std. dev., size (samples)
data2 = np.random.normal(100, 13, 500)  # mean, std. dev., size (samples)

statistic, pvalue = ks_2samp(data1, data2)
if pvalue < 0.05:
    print("The sample distributions are different (reject null hypothesis)")
else:
    print("The sample distributions are the same (do not reject null hypothesis)")
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))

The sample distributions are different (reject null hypothesis)
The statistic is: 0.094, the p-value is: 0.02406614705241275

Kruskal-Wallis Test

The Kruskal Wallis test is a nonparametric test that compares three or more independent groups. It can be thought of as a one-way ANOVA on ranks.

from scipy.stats import kruskal

group1 = [7, 14, 14, 13, 12, 9, 6, 14, 12, 8]
group2 = [15, 17, 13, 15, 15, 13, 9, 12, 10, 8]
group3 = [6, 8, 8, 9, 5, 14, 13, 8, 10, 9]

statistic, pvalue = kruskal(group1, group2, group3)
if pvalue >= 0.05:
    print("There is no significant difference between the groups.")
else:
    print("There is a significant difference between the groups.")
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))

There is a significant difference between the groups.
The statistic is: 6.287801578353988, the p-value is: 0.043114289703508814

Kurtosis

Kurtosis is a measure of the 'tailedness' of a distribution.

data = np.random.normal(loc=20, scale=5, size=70)

distplot = sns.distplot(data, color='green')

from scipy.stats import skew
skew(data)

np.float64(0.3419056042636773)

Levenes's Test

This test determines if variances are equal between measures.

from scipy.stats import levene

measure1 = np.random.normal(100, 30, 300) # mean 100, std. dev 30, 300 samples
measure2 = np.random.normal(100, 60, 300) # mean 100, std. dev 60, 300 samples

# Plot data
plt.figure(figsize=(8,5))
plt.boxplot([measure1, measure2], labels=['Measure 1', 'Measure 2'])
plt.show()

# Perform Levene test
statistic, pvalue = levene(measure1, measure2)
if pvalue >= 0.05:
    print("There is no significant difference between the measures.")
else:
    print("There is a significant difference between the measures.")
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))

There is a significant difference between the measures.
The statistic is: 106.37300499889395, the p-value is: 4.5767897769439415e-23

Logrank Test (and Kaplan–Meier curve)

This test compares the survival distributions of two independent groups.

from sksurv.nonparametric import kaplan_meier_estimator
from sksurv.compare import compare_survival

# Your data
df = pd.DataFrame({
    'durations': [4, 11, 13, 16, 12, 19, 6, 20, 18, 21],
    'events': [1, 0, 1, 1, 1, 1, 1, 1, 1, 1],
    'drug_group': ['A', 'A', 'A', 'A', 'A', 'B', 'B', 'B', 'B', 'B']
})

# Combine all data into one structured array
y = np.array([(bool(e), t) for e, t in zip(df['events'], df['durations'])],
             dtype=[('event', bool), ('time', float)])

# Group indicator - which group each observation belongs to
group_indicator = df['drug_group'].values

# Perform Log-rank test
chisq, pvalue = compare_survival(y, group_indicator)

if pvalue >= 0.05:
    print("The distribution curves for each group are the same.")
else:
    print("The distribution curves for each group are not the same.")
print(f"Chi-square statistic: {chisq:.4f}, p-value: {pvalue:.4f}")

# Kaplan-Meier visualisation
drug_a = df[df['drug_group'] == 'A']
drug_b = df[df['drug_group'] == 'B']

plt.figure(figsize=(8, 5))
# Plot Drug A
time_a, survival_a = kaplan_meier_estimator(
    drug_a['events'].astype(bool).values, 
    drug_a['durations'].values
)
plt.step(time_a, survival_a, where="post", label="Drug A", color='blue')

# Plot Drug B
time_b, survival_b = kaplan_meier_estimator(
    drug_b['events'].astype(bool).values, 
    drug_b['durations'].values
)
plt.step(time_b, survival_b, where="post", label="Drug B", color='green')
plt.xlabel("Time")
plt.ylabel("Survival Probability")
plt.title("Kaplan–Meier Survival Curves by Drug Group")
plt.legend()
plt.grid(True)
plt.show()

The distribution curves for each group are not the same.
Chi-square statistic: 3.8997, p-value: 0.0483

Mann–Whitney U test

This test determines whether the mean values of two independent groups differ significantly from each other.

from scipy.stats import mannwhitneyu

group1 = [12, 15, 14, 10, 9, 13]
group2 = [18, 20, 17, 22, 19, 11]

statistic, pvalue = mannwhitneyu(group1, group2, alternative='two-sided')
if pvalue >= 0.05:
    print("There is no significant difference between the measures.")
else:
    print("There is a significant difference between the measures.")
    
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))

There is a significant difference between the measures.
The statistic is: 4.0, the p-value is: 0.025974025974025976

Mauchly's Test of Sphericity

The Mauchly's test of sphericity is used to assess whether or not the assumption of sphericity is met. The assumption of sphericity is met when there is equal variance of difference scores across different measurement periods.

from pingouin import sphericity

df = pd.DataFrame({'A': [65, 92, 28, 68, 24],
                     'B': [58, 48, 55, 60, 45],
                     'C': [60, 49, 55, 64, 49]})

pvalue = round(sphericity(df, method='mauchly')[-1], 3)
if pvalue >= 0.05:
    print("Variances of the differences are equal.")
else:
    print("Variances of the differences are not equal")
print("The p-value is: " + str(pvalue))

Variances of the differences are not equal
The p-value is: 0.001

One Proportion Z-test

A one proportion z-test is used to compare an observed proportion to a theoretical one.

from statsmodels.stats.proportion import proportions_ztest

successes = 50
trials = 200
observed_proportion = successes/trials

theoretical_proportion = 0.12   # 12 %

statistic, pvalue = proportions_ztest(count=successes, nobs=trials, value=theoretical_proportion)

if pvalue < 0.05:
    print("Reject the null hypothesis: the observed proportion (" + str(observed_proportion) + ") differs from the theoretical proportion (" + str(theoretical_proportion) +").")
else:
    print("Fail to reject the null hypothesis: the observed proportion (" + str(observed_proportion) + ") does not differ from the theoretical proportion (" + str(theoretical_proportion) +").")

Reject the null hypothesis: the observed proportion (0.25) differs from the theoretical proportion (0.12).

One-Sample Cramér-von Mises Test

This test determines whether a data set is normally distributed.

from scipy.stats import cramervonmises

sample = random.sample(range(0, 99), 80) # assign 80 random (not normally distributed) to list

result = cramervonmises(sample, 'norm')
statistic = result.statistic
pvalue = result.pvalue

if pvalue >= 0.05:
    print("Data is likely normally distributed.")
else:
    print("Data is likely not normally distributed.")
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))

Data is likely not normally distributed.
The statistic is: 25.590586550519497, the p-value is: 4.37574143585806e-09

Pearson R Correlation

Pearson's correlation coefficient is a measure of the strength of a linear association between two variables.

from scipy.stats import pearsonr

kg = [85, 95, 98, 86, 67, 83, 78, 64, 79, 87, 60, 86, 99, 73, 94, 88, 83, 68, 99, 75, 98, 68, 93]
calories = [2465, 2945, 2940, 2838, 2010, 2407, 2418, 1856, 2607, 2871, 1920, 2752, 2970, 2263, 3102, 2816, 2739, 1972, 3168, 2400, 3234, 2040, 2976]


statistic, pvalue = pearsonr(kg, calories) 
if pvalue >= 0.05:
    print("There is a correlation between the values.")
else:
    print("There is not a correlation between the values.")
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))

There is not a correlation between the values.
The statistic is: 0.9619689837547748, the p-value is: 2.604391095408307e-13

Skewness

Skewness is a measure of the asymmetry of a distribution.

from scipy.stats import skew

rng = np.random.default_rng(6)
data = rng.normal(loc=20, scale=5, size=70)

distplot = sns.distplot(data, color='green')

skew_value = skew(data)

if skew_value > 0:
    print("Data in positively skewed, skewness value: " + str(round(skew_value,3)))
elif skew_value < 0:
    print("Data in negatively skewed, skewness value: " + str(round(skew_value,3)))
else:
    print("There is no skew (skewness value is 0)")

Data in negatively skewed, skewness value: -0.361

Shapiro-Wilk

This test determines whether a data set is normally distributed.

from scipy.stats import shapiro

data = np.random.normal(loc = 20, scale = 5, size=150) # mean 20, std. dev 5, 150 samples

statistic, pvalue= shapiro(data)

if pvalue >= 0.05:
    print("Data is normally distibuted.")
else:
    print("Data is not normally distibuted.")
    
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue)) 

Data is normally distibuted.
The statistic is: 0.9843264017669557, the p-value is: 0.08656599192770058

T-test

Independent (Student)

This test compares the means of two independent groups (assuming equal variances).

from scipy.stats import ttest_ind

height_female = np.random.normal(loc=170, scale=6, size=10) # mean 170, std. dev 6, 10 samples
height_male = np.random.normal(loc=163, scale=5, size=10) # mean 163, std. dev 5, 10 samples

statistic, pvalue = ttest_ind(height_female, height_male)
if pvalue >= 0.05:
    print("There is no significant difference between the means.")
else:
    print("There is a significant difference between the means.")
    
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))    

There is no significant difference between the means.
The statistic is: 0.5124084680416063, the p-value is: 0.6145954059634584

Independent (Welch)

This test compares the means of two independent groups (without assuming equal variances).

from scipy.stats import ttest_ind

height_female = [175, 174, 174, 168, 166, 169, 172, 182, 169, 171, 175]
height_male = [181, 183, 178, 178, 171, 169, 173, 191, 168, 174, 177]

statistic, pvalue = ttest_ind(height_female, height_male, equal_var = False)

if pvalue >= 0.05:
    print("There is no significant difference between the groups.")
else:
    print("There is a significant difference between the groups.")
    
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))   

There is no significant difference between the groups.
The statistic is: -1.7940948120062388, the p-value is: 0.09031762651932743

Related

This test compares two means where one sample can be paired with observations in another sample.

from scipy.stats import ttest_rel

before = [20, 17, 12, 25, 21, 23, 19, 15, 15, 16, 22]
after = [22, 19, 17, 26, 21, 25, 25, 20, 18, 15, 23]

statistic, pvalue = ttest_rel(before, after)

if pvalue >= 0.05:
    print("There is no significant difference between the groups.")
else:
    print("There is a significant difference between the groups.")
    
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))   

There is a significant difference between the groups.
The statistic is: -3.557973584775827, the p-value is: 0.005198756772396905

Two Proportion Z-test

A two-proportions z-test compares the proportions of two independent samples.

from statsmodels.stats.proportion import proportions_ztest

successes = [180, 230]   # number of conversions in A and B
totals    = [1000, 950]  # total visitors in A and B

# Two-proportion z-test
statistic, pvalue = proportions_ztest(successes, totals, alternative='two-sided')

if pvalue < 0.05:
    print("There is a significant difference between the groups.")
else:
    print("There is not a significant difference between the groups.")

There is a significant difference between the groups.

Wilcoxon Sign-Ranked Test

This test determines whether the mean values of two dependent groups differ significantly from each other.

from scipy.stats import wilcoxon

measure1 = [14.3, 13.1, 12.1, 12.6, 9.5, 12.6, 8.1, 8.9, 9.0, 8.3, 7.9, 8.1, 13.4]
measure2 = [12.7, 11.1, 15.3, 12.7, 10.5, 15.6, 11.2, 14.2, 16.3, 15.5, 19.9, 20.4, 9.8]

statistic, pvalue = wilcoxon(measure1, measure2)
if pvalue >= 0.05:
    print("There is no significant difference between the measures.")
else:
    print("There is a significant difference between the measures.")
    
print("The statistic is: " + str(statistic) + ", the p-value is: " + str(pvalue))

There is a significant difference between the measures.
The statistic is: 15.0, the p-value is: 0.03271484375