Chi-squared Distribution: Worked Example

Chi-squared DistributionWorked Example

📊 Understanding the Chi‑Squared Test: A Complete Worked Example
The Chi‑Squared (χ²) test is a simple but powerful method for analysing categorical data. It helps determine whether differences in your data are meaningful or simply due to chance.

🔍 What Does the Chi‑Squared Test Do?
The Chi‑Squared test compares:

what you observed, and
what you would expect if there were no relationship
The test statistic is:

$\chi ^2=\sum \frac{(O-E)^2}{E}$
Where O = observed frequency and E = expected frequency.

📁 Scenario
You want to test whether DNA (Did Not Attend) rates are related to age group in an outpatient clinic.

📋 Observed Data

Age Group	Attended	DNA	Total
Under 40	40	20	60
40 and over	30	10	40
Total	70	30	100

1️⃣ Hypotheses

Null hypothesis H_0: DNA is independent of age group
Alternative hypothesis H_1: DNA is not independent of age group

2️⃣ Observed Matrix
$O=\left( \begin{matrix}40&20\\ 30&10\end{matrix}\right)$

3️⃣ Expected Frequencies
Expected values use:
$E_{ij}=\frac{(\mathrm{Row\ Total})_i\cdot (\mathrm{Column\ Total})_j}{\mathrm{Grand\ Total}}$

Calculations
$E_{11}=\frac{60\times 70}{100}=42$
$E_{12}=\frac{60\times 30}{100}=18$
$E_{21}=\frac{40\times 70}{100}=28$
$E_{22}=\frac{40\times 30}{100}=12$

📋 Expected Frequencies Table

Age Group	Attended (E)	DNA (E)	Total
Under 40	42	18	60
40 and over	28	12	40
Total	70	30	100

4️⃣ Chi‑Squared Calculation
Formula:

$\chi ^2=\sum \frac{(O-E)^2}{E}$

Cell contributions
$\frac{(40-42)^2}{42}=0.0952$

$\frac{(20-18)^2}{18}=0.2222$
$\frac{(30-28)^2}{28}=0.1429$
$\frac{(10-12)^2}{12}=0.3333$

Total Chi‑Squared
$\chi ^2=0.7936$

5️⃣ Degrees of Freedom

For an (r\times c) table:
$df=(r-1)(c-1)$
Here:
r=2
c=2
So:

6️⃣ Decision
Critical value at df=1, \alpha =0.05:
$\chi _{\mathrm{critical}}^2=3.84$

$\chi _{\mathrm{calc}}^2=0.79$
Since 0.79<3.84, we fail to reject the null hypothesis.

7️⃣ Interpretation
There is no statistically significant association between age group and DNA status**.
Any differences could reasonably be due to chance.

🎯 Final Thoughts
This example shows the full workflow of a Chi‑Squared Test of Independence:

Clean data table
Observed vs expected values
Chi‑Squared calculation
Degrees of freedom
Statistical decision

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