Goodness of fit evaluation in SciPy - Step-by-Step Execution

Concept Flow - Goodness of fit evaluation

Collect observed data

↓

Define expected distribution

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Calculate test statistic

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Compare statistic to distribution

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Get p-value

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Decide if fit is good or not

We start with observed data and an expected distribution, calculate a test statistic, then find a p-value to decide if the data fits well.

Execution Sample

SciPy

from scipy.stats import chisquare
observed = [16, 18, 16, 14, 12, 12]
expected = [15, 15, 15, 15, 15, 15]
stat, p = chisquare(f_obs=observed, f_exp=expected)
print(stat, p)

This code runs a chi-square goodness of fit test comparing observed counts to expected counts.

Execution Table

Step	Action	Calculation	Result
1	Calculate differences (observed - expected)	[16-15, 18-15, 16-15, 14-15, 12-15, 12-15]	[1, 3, 1, -1, -3, -3]
2	Square differences	[1^2, 3^2, 1^2, (-1)^2, (-3)^2, (-3)^2]	[1, 9, 1, 1, 9, 9]
3	Divide squared differences by expected	[1/15, 9/15, 1/15, 1/15, 9/15, 9/15]	[0.0667, 0.6, 0.0667, 0.0667, 0.6, 0.6]
4	Sum all values	0.0667+0.6+0.0667+0.0667+0.6+0.6	2.0
5	Calculate p-value from chi-square distribution with df=5	p = 1 - CDF(2.0, df=5)	p = 0.849
6	Decision	p > 0.05 means fit is good	Fail to reject null hypothesis

💡 Test ends after p-value calculation and decision step

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	After Step 5	Final
observed	[16,18,16,14,12,12]	[16,18,16,14,12,12]	[16,18,16,14,12,12]	[16,18,16,14,12,12]	[16,18,16,14,12,12]	[16,18,16,14,12,12]	[16,18,16,14,12,12]
expected	[15,15,15,15,15,15]	[15,15,15,15,15,15]	[15,15,15,15,15,15]	[15,15,15,15,15,15]	[15,15,15,15,15,15]	[15,15,15,15,15,15]	[15,15,15,15,15,15]
diff	N/A	[1,3,1,-1,-3,-3]	[1,3,1,-1,-3,-3]	[1,3,1,-1,-3,-3]	[1,3,1,-1,-3,-3]	[1,3,1,-1,-3,-3]	[1,3,1,-1,-3,-3]
squared_diff	N/A	N/A	[1,9,1,1,9,9]	[1,9,1,1,9,9]	[1,9,1,1,9,9]	[1,9,1,1,9,9]	[1,9,1,1,9,9]
chi_components	N/A	N/A	N/A	[0.0667,0.6,0.0667,0.0667,0.6,0.6]	[0.0667,0.6,0.0667,0.0667,0.6,0.6]	[0.0667,0.6,0.0667,0.0667,0.6,0.6]	[0.0667,0.6,0.0667,0.0667,0.6,0.6]
statistic	N/A	N/A	N/A	N/A	2.0	2.0	2.0
p_value	N/A	N/A	N/A	N/A	N/A	0.849	0.849

Key Moments - 3 Insights

Why do we square the differences between observed and expected counts?

Squaring makes all differences positive and emphasizes larger differences, as shown in step 2 of the execution_table.

What does a high p-value mean in this test?

A high p-value (like 0.849 in step 5) means the observed data fits the expected distribution well, so we do not reject the fit.

Why do we divide squared differences by expected counts?

Dividing by expected counts normalizes the differences, so categories with larger expected counts don't dominate, as in step 3.

Visual Quiz - 3 Questions

Look at the execution_table at step 4, what is the sum of the chi-square components?

A 2.0

B 1.5

C 3.0

D 0.85

💡 Hint

Check the 'Sum all values' calculation in step 4 of the execution_table.

According to variable_tracker, what is the value of p_value after step 5?

A 0.15

B 0.849

C 0.05

D 2.0

💡 Hint

Look at the 'p_value' row under 'After Step 5' in variable_tracker.

If the observed counts were all equal to expected counts, what would the chi-square statistic be?

A 6

B 1

C 0

D 15

💡 Hint

When observed equals expected, differences are zero, so sum of squared differences is zero (see step 1 and 2).

Concept Snapshot

Goodness of fit test compares observed data to expected distribution.
Calculate chi-square statistic: sum((observed - expected)^2 / expected).
Find p-value from chi-square distribution with degrees of freedom = categories - 1.
High p-value means data fits well; low p-value means poor fit.
Use scipy.stats.chisquare for easy calculation.

Full Transcript

Goodness of fit evaluation checks if observed data matches an expected pattern. We start by collecting observed counts and defining expected counts. Then, we calculate the differences between observed and expected, square them, and divide by expected counts. Summing these gives the chi-square statistic. Using the chi-square distribution with the right degrees of freedom, we find the p-value. A high p-value means the observed data fits the expected distribution well. This process is shown step-by-step in the execution table and variable tracker. Key points include squaring differences to avoid negatives, normalizing by expected counts, and interpreting the p-value correctly.