To calculate the range and variance for the data shown in the table, we need to follow these steps:

1. Range Calculation:

The range is the difference between the maximum and minimum values of the midpoints of the class intervals.

• The lowest class is 10–14, and the midpoint ($x$) is 12.
• The highest class is 35–39, and the midpoint ($x$) is 37.

Range = $\text{Max} \, x - \text{Min} \, x = 37 - 12 = 25$

2. Variance Calculation:

The formula for variance $\sigma^2$ for grouped data is:

$\sigma^2 = \frac{\sum f(x-\bar{x})^2}{N}$

Where:

• $f$ is the frequency (No. of students).
• $x$ is the midpoint.
• $\bar{x}$ is the mean.
• $N$ is the total number of students.

Step 1: Calculate Mean $\bar{x}$

We can calculate the mean using the formula:

$\bar{x} = \frac{\sum fx}{N}$

From the table: $\sum fx = 36 + 204 + 418 + 972 + 704 + 296 = 2630$ $N = 3 + 12 + 19 + 36 + 22 + 8 = 100$

Thus, the mean $\bar{x}$ is:

$\bar{x} = \frac{2630}{100} = 26.3$

Step 2: Calculate Variance

Now, sum up the column $f(x-\bar{x})^2$:

$\sum f(x-\bar{x})^2 = 613.47 + 1037.88 + 351.31 + 17.64 + 714.78 + 915.92 = 3650$

Now, plug the values into the variance formula:

$\sigma^2 = \frac{3650}{100} = 36.5$

Final Answers:

• Range: 25
• Variance: 36.5

Would you like a breakdown of the steps or have any further questions?

5 Related Questions:

1. How do you calculate the standard deviation from the variance?
2. What is the significance of variance in data analysis?
3. How does the class width affect the range and variance of the dataset?
4. What if the class intervals were not equal? How would the calculation change?
5. Can you calculate the coefficient of variation for this data?

Tip:

The standard deviation is simply the square root of the variance, which provides a measure of how spread out the data is in the same units as the original data.