By definition of the mean, the following relationship must hold: The sum of all values in the data must equal n x mean, where n is the number of values in the data set.
So if a data set has 10 values, the sum of the 10 values must equal the mean x 10.
If the mean of the 10 values is 3.
5 (you could pick any number), this constraint requires that the sum of the 10 values must equal 10 x 3.
5 = 35.
With that constraint, the first value in the data set is free to vary.
Whatever value it is, it’s still possible for the sum of all 10 numbers to have a value of 35.
The second value is also free to vary, because whatever value you choose, it still allows for the possibility that the sum of all the values is 35.
Now Let’s see some of widely used hypothesis testing type :-T Test ( Student T test)Z TestANOVA TestChi-Square TestT- Test :- A t-test is a type of inferential statistic which is used to determine if there is a significant difference between the means of two groups which may be related in certain features.
It is mostly used when the data sets, like the set of data recorded as outcome from flipping a coin a 100 times, would follow a normal distribution and may have unknown variances.
T test is used as a hypothesis testing tool, which allows testing of an assumption applicable to a population.
T-test has 2 types : 1.
one sampled t-test 2.
two-sampled t-test.
One sample t-test : The One Sample t Test determines whether the sample mean is statistically different from a known or hypothesised population mean.
The One Sample t Test is a parametric test.
Example :- you have 10 ages and you are checking whether avg age is 30 or not.
(check code below for that using python)from scipy.
stats import ttest_1sampimport numpy as npages = np.
genfromtxt(“ages.
csv”)print(ages)ages_mean = np.
mean(ages)print(ages_mean)tset, pval = ttest_1samp(ages, 30)print(“p-values”,pval)if pval < 0.
05: # alpha value is 0.
05 or 5% print(" we are rejecting null hypothesis")else: print("we are accepting null hypothesis")Output for above code is :one-sample t-test outputTwo sampled T-test :-The Independent Samples t Test or 2-sample t-test compares the means of two independent groups in order to determine whether there is statistical evidence that the associated population means are significantly different.
The Independent Samples t Test is a parametric test.
This test is also known as: Independent t Test.
Example : is there any association between week1 and week2 ( code is given below in python)from scipy.
stats import ttest_indimport numpy as npweek1 = np.
genfromtxt("week1.
csv", delimiter=",")week2 = np.
genfromtxt("week2.
csv", delimiter=",")print(week1)print("week2 data :-.")print(week2)week1_mean = np.
mean(week1)week2_mean = np.
mean(week2)print("week1 mean value:",week1_mean)print("week2 mean value:",week2_mean)week1_std = np.
std(week1)week2_std = np.
std(week2)print("week1 std value:",week1_std)print("week2 std value:",week2_std)ttest,pval = ttest_ind(week1,week2)print("p-value",pval)if pval <0.
05: print("we reject null hypothesis")else: print("we accept null hypothesis")2-sampled t-test outputPaired sampled t-test :- The paired sample t-test is also called dependent sample t-test.
It’s an uni variate test that tests for a significant difference between 2 related variables.
An example of this is if you where to collect the blood pressure for an individual before and after some treatment, condition, or time point.
H0 :- means difference between two sample is 0H1:- mean difference between two sample is not 0check the code below for sameimport pandas as pdfrom scipy import statsdf = pd.
read_csv("blood_pressure.
csv")df[['bp_before','bp_after']].
describe()ttest,pval = stats.
ttest_rel(df['bp_before'], df['bp_after'])print(pval)if pval<0.
05: print("reject null hypothesis")else: print("accept null hypothesis")When you can run a Z Test.
Several different types of tests are used in statistics (i.
e.
f test, chi square test, t test).
You would use a Z test if:Your sample size is greater than 30.
Otherwise, use a t test.
Data points should be independent from each other.
In other words, one data point isn’t related or doesn’t affect another data point.
Your data should be normally distributed.
However, for large sample sizes (over 30) this doesn’t always matter.
Your data should be randomly selected from a population, where each item has an equal chance of being selected.
Sample sizes should be equal if at all possible.
Example again we are using z-test for blood pressure with some mean like 156 (python code is below for same) one-sample Z test.
import pandas as pdfrom scipy import statsfrom statsmodels.
stats import weightstats as stestsztest ,pval = stests.
ztest(df['bp_before'], x2=None, value=156)print(float(pval))if pval<0.
05: print("reject null hypothesis")else: print("accept null hypothesis")Two-sample Z test- In two sample z-test , similar to t-test here we are checking two independent data groups and deciding whether sample mean of two group is equal or not.
H0 : mean of two group is 0H1 : mean of two group is not 0Example : we are checking in blood data after blood and before blood data.
(code in python below)ztest ,pval1 = stests.
ztest(df['bp_before'], x2=df['bp_after'], value=0,alternative='two-sided')print(float(pval1))if pval<0.
05: print("reject null hypothesis")else: print("accept null hypothesis")ANOVA (F-TEST) :- The t-test works well when dealing with two groups, but sometimes we want to compare more than two groups at the same time.
For example, if we wanted to test whether voter age differs based on some categorical variable like race, we have to compare the means of each level or group the variable.
We could carry out a separate t-test for each pair of groups, but when you conduct many tests you increase the chances of false positives.
The analysis of variance or ANOVA is a statistical inference test that lets you compare multiple groups at the same time.
F = Between group variability / Within group variabilityF-Test or Anova concept imageUnlike the z and t-distributions, the F-distribution does not have any negative values because between and within-group variability are always positive due to squaring each deviation.
One Way F-test(Anova) :- It tell whether two or more groups are similar or not based on their mean similarity and f-score.
Example : there are 3 different category of plant and their weight and need to check whether all 3 group are similar or not (code in python below)df_anova = pd.
read_csv('PlantGrowth.
csv')df_anova = df_anova[['weight','group']]grps = pd.
unique(df_anova.
group.
values)d_data = {grp:df_anova['weight'][df_anova.
group == grp] for grp in grps} F, p = stats.
f_oneway(d_data['ctrl'], d_data['trt1'], d_data['trt2'])print("p-value for significance is: ", p)if p<0.
05: print("reject null hypothesis")else: print("accept null hypothesis")Two Way F-test :- Two way F-test is extension of 1-way f-test, it is used when we have 2 independent variable and 2+ groups.
2-way F-test does not tell which variable is dominant.
if we need to check individual significance then Post-hoc testing need to be performed.
Now let’s take a look at the Grand mean crop yield (the mean crop yield not by any sub-group), as well the mean crop yield by each factor, as well as by the factors grouped togetherimport statsmodels.
api as smfrom statsmodels.
formula.
api import olsdf_anova2 = pd.
read_csv("https://raw.
githubusercontent.
com/Opensourcefordatascience/Data-sets/master/crop_yield.
csv")model = ols('Yield ~ C(Fert)*C(Water)', df_anova2).
fit()print(f"Overall model F({model.
df_model: .
0f},{model.
df_resid: .
0f}) = {model.
fvalue: .
3f}, p = {model.
f_pvalue: .
4f}")res = sm.
stats.
anova_lm(model, typ= 2)resChi-Square Test- The test is applied when you have two categorical variables from a single population.
It is used to determine whether there is a significant association between the two variables.
For example, in an election survey, voters might be classified by gender (male or female) and voting preference (Democrat, Republican, or Independent).
We could use a chi-square test for independence to determine whether gender is related to voting preferencecheck example in python belowdf_chi = pd.
read_csv('chi-test.
csv')contingency_table=pd.
crosstab(df_chi["Gender"],df_chi["Shopping?"])print('contingency_table :-.',contingency_table)#Observed ValuesObserved_Values = contingency_table.
values print("Observed Values :-.",Observed_Values)b=stats.
chi2_contingency(contingency_table)Expected_Values = b[3]print("Expected Values :-.",Expected_Values)no_of_rows=len(contingency_table.
iloc[0:2,0])no_of_columns=len(contingency_table.
iloc[0,0:2])ddof=(no_of_rows-1)*(no_of_columns-1)print("Degree of Freedom:-",ddof)alpha = 0.
05from scipy.
stats import chi2chi_square=sum([(o-e)**2.
/e for o,e in zip(Observed_Values,Expected_Values)])chi_square_statistic=chi_square[0]+chi_square[1]print("chi-square statistic:-",chi_square_statistic)critical_value=chi2.
ppf(q=1-alpha,df=ddof)print('critical_value:',critical_value)#p-valuep_value=1-chi2.
cdf(x=chi_square_statistic,df=ddof)print('p-value:',p_value)print('Significance level: ',alpha)print('Degree of Freedom: ',ddof)print('chi-square statistic:',chi_square_statistic)print('critical_value:',critical_value)print('p-value:',p_value)if chi_square_statistic>=critical_value: print("Reject H0,There is a relationship between 2 categorical variables")else: print("Retain H0,There is no relationship between 2 categorical variables") if p_value<=alpha: print("Reject H0,There is a relationship between 2 categorical variables")else: print("Retain H0,There is no relationship between 2 categorical variables")You can get all code in my git repository.
ah, finally we came to end of this article.
I hope this article would have helped.
any feedback is always appreciated.
For more update check my git and follow we on medium.
.. More details