Comparing Age Distributions: 2000 Census vs. 2010 Sample and Chi-Square Test Analysis

To understand the age distribution in the 2010 sample in the context of the 2000 Census data, you can compare the proportions from both datasets. Here's how you can interpret it:

1. **Proportions in 2000 Census:**

   - Age 18: 20%

   - Age 18-35: 30%

   - Age >35: 50%

2. **Proportions in 2010 Sample:**

   - Age 18: 121 out of 500 = 24.2%

   - Age 18-35: 288 out of 500 = 57.6%

   - Age >35: 91 out of 500 = 18.2%

**Analysis:**

1. **Age 18:** 

   - **2000 Census:** 20%

   - **2010 Sample:** 24.2%

   - The percentage of people aged 18 in the 2010 sample is higher compared to the 2000 Census, indicating a relative increase.

2. **Age 18-35:**

   - **2000 Census:** 30%

   - **2010 Sample:** 57.6%

   - The proportion of people aged 18-35 in the 2010 sample is significantly higher compared to the 2000 Census, suggesting a shift toward a younger demographic.

3. **Age >35:**

   - **2000 Census:** 50%

   - **2010 Sample:** 18.2%

   - The proportion of people older than 35 in the 2010 sample is much lower than in the 2000 Census, indicating a decline in this age group.

**Summary:** The 2010 sample shows a significant shift towards a younger age distribution compared to the 2000 Census data, with a higher percentage of individuals aged 18-35 and a lower percentage aged over 35. This could reflect changes in population demographics, migration patterns, or other social trends between 2000 and 2010.

**Chi-square Test Analysis:**

1. **Create a Contingency Table:**

   - **Observed frequencies (from 2010 sample):**

     - Age 18: 121

     - Age 18-35: 288

     - Age >35: 91

   - **Expected frequencies (based on 2000 Census proportions):**

     - Total sample size in 2010: 500

     - Expected frequency for Age 18: 20% of 500 = 100

     - Expected frequency for Age 18-35: 30% of 500 = 150

     - Expected frequency for Age >35: 50% of 500 = 250

2. **Calculate the Chi-square statistic:**

   Use the formula:

   

   χ² = Σ ((O_i - E_i)² / E_i)

   

   Where `O_i` is the observed frequency and `E_i` is the expected frequency.

   For each age group:

   - Age 18: `((121 - 100)² / 100) = (21² / 100) = 4.41`

   - Age 18-35: `((288 - 150)² / 150) = (138² / 150) ≈ 127.4`

   - Age >35: `((91 - 250)² / 250) = (159² / 250) ≈ 101.7`

   Total χ² value = `4.41 + 127.4 + 101.7 = 233.51`

3. **Determine the degrees of freedom:**

   Degrees of freedom `df = (number of categories - 1)`

   Here, `df = 3 - 1 = 2`

4. **Compare the Chi-square statistic to the critical value from the Chi-square distribution table** or compute the p-value to determine statistical significance.

If the Chi-square statistic is greater than the critical value (or if the p-value is less than the significance level, usually 0.05), then you can conclude that there is a significant difference between the distributions.

**Observed Value vs. Expected Value:**

- **Observed Value:** This is the number or frequency you actually observed in each category. For example, in the 2010 sample, the observed values were:

  - Age 18: 121

  - Age 18-35: 288

  - Age >35: 91

- **Expected Value:** This is the number or frequency you would expect if the proportions from the 2000 Census data applied to the 2010 sample. Based on the 2000 Census data:

  - For Age 18: 20% of 500 = 100

  - For Age 18-35: 30% of 500 = 150

  - For Age >35: 50% of 500 = 250

In a Chi-square test, the difference between the observed value and the expected value is used to determine if there is a significant deviation from what would be expected. This difference is squared and divided by the expected value to calculate the Chi-square statistic.

**Chi-square Test Suitability:**

1. **Categorical Data Suitability:** The Chi-square test is designed for categorical data, which can be either nominal (e.g., types of fruits) or ordinal (e.g., levels of education). It assesses the relationship between categorical variables and whether the observed distribution differs from what would be expected by chance.

2. **Goodness-of-Fit Test:** It checks if the observed frequencies of categories match an expected distribution based on a theoretical model or historical data. For example, it can test if the age distribution in a sample matches the distribution expected based on past census data.

3. **Independence Test:** It evaluates whether two categorical variables are independent of each other. For instance, it can test if age and preference for a particular product are related or independent.

4. **Distribution of Data:** The Chi-square test is based on the idea that the difference between observed and expected frequencies (when squared and normalized) follows a Chi-square distribution. This allows for hypothesis testing regarding the distribution of data across categories.

5. **Assumptions and Flexibility:** The test assumes that data are independent and that sample sizes are large enough to approximate the Chi-square distribution. It is flexible and doesn’t require assumptions about the underlying data distribution, making it useful for various practical situations.

Overall, the Chi-square test is a robust method for analyzing categorical data and determining if differences in observed counts can be attributed to chance or if they indicate a statistically significant effect.

Understanding z scores and standard deviation

### **Visualizing Age Distribution by Gender in Python**  

When analyzing demographic data, one of the most insightful visualizations is an **age distribution plot** categorized by **gender**. This helps us understand trends, such as whether one gender tends to be older or younger in the dataset.

#### **Step 1: Import Necessary Libraries**  

Before we start, let's import the required Python libraries:

import pandas as pd

import matplotlib.pyplot as plt

import seaborn as sns

#### **Step 2: Sample Data Preparation**  

Let's assume we have a dataset stored in a **Pandas DataFrame** with two columns: `age` and `gender`. If you are working with real-world data, you can load it using `pd.read_csv()`.

# Creating a sample dataset

data = {'age': [25, 30, 22, 35, 40, 28, 50, 19, 60, 45, 32, 37, 29, 23, 41],

        'gender': ['Male', 'Female', 'Male', 'Female', 'Male', 'Female', 'Male', 

                   'Female', 'Male', 'Female', 'Male', 'Female', 'Male', 'Female', 'Male']}

df = pd.DataFrame(data)

#### **Step 3: Plot Age Distribution for Each Gender**  

We will use **seaborn’s** `histplot()` to create histograms for each gender. This will allow us to see the age distribution separately for males and females.

plt.figure(figsize=(10, 6))

# Creating a histogram for age distribution by gender

sns.histplot(data=df, x='age', hue='gender', kde=True, bins=10, alpha=0.6)

# Adding labels and title

plt.xlabel('Age')

plt.ylabel('Count')

plt.title('Age Distribution by Gender')

plt.legend(title='Gender')

# Show the plot

plt.show()

#### **Explanation of the Code**

1. **`sns.histplot()`** is used to plot a histogram of ages.

2. **`hue='gender'`** colors the bars based on gender.

3. **`kde=True`** adds a density curve to visualize distribution smoothly.

4. **`bins=10`** divides the age range into 10 intervals.

5. **`alpha=0.6`** makes bars slightly transparent for better visualization.

#### **Alternative: Using Boxplot for Distribution Comparison**

If you want to compare age distribution between genders in a different way, a **boxplot** is another great option:

plt.figure(figsize=(8, 6))

sns.boxplot(x='gender', y='age', data=df, palette='coolwarm')

# Adding labels and title

plt.xlabel('Gender')

plt.ylabel('Age')

plt.title('Age Distribution by Gender')

plt.show()

#### **Conclusion**

- **Histograms** are useful for visualizing the frequency of different ages within each gender.

- **Boxplots** help understand the **median age, range, and outliers** for each gender.

- Using both together provides a **comprehensive view** of age distribution.

If you have a **large dataset**, consider using **violin plots** or **strip plots** for deeper insights.