Calculate the standard deviation of the following data: 7, 8, 9, 10, 11, 12, 13

To calculate the standard deviation of a set of data, we follow these systematic steps:

1. Analyze the Question

We are given a set of seven numbers: 7, 8, 9, 10, 11, 12, 13. We need to find the standard deviation ($\sigma$), which measures the amount of variation or dispersion in the set.

2. Explain the Concept

The formula for standard deviation is:

Where:

• $x$ = each value in the data set
• $\bar{x}$ = the mean (average) of the data
• $n$ = the number of values

3. Show the Solution

Step 1: Calculate the mean ($\bar{x}$)

Step 2: Subtract the mean from each value and square the result $(x - \bar{x})^2$

• $(7 - 10)^2 = (-3)^2 = 9$
• $(8 - 10)^2 = (-2)^2 = 4$
• $(9 - 10)^2 = (-1)^2 = 1$
• $(10 - 10)^2 = (0)^2 = 0$
• $(11 - 10)^2 = (1)^2 = 1$
• $(12 - 10)^2 = (2)^2 = 4$
• $(13 - 10)^2 = (3)^2 = 9$

Step 3: Find the sum of these squares

Step 4: Divide by the number of values ($n$) and take the square root

4. Identify the Answer

Comparing our result to the options provided, the correct answer is A.

5. Exam Tip

When you see a set of numbers that are in an arithmetic progression (like this one, where the difference between consecutive numbers is constant, i.e., 1), you can quickly calculate the variance using the formula $\frac{n^2 - 1}{12} \times d^2$, where $n$ is the number of terms and $d$ is the common difference. Here, $n=7$ and $d=1$: Variance $= \frac{7^2 - 1}{12} \times 1^2 = \frac{48}{12} = 4$. Standard Deviation $= \sqrt{4} = 2$. This is a great shortcut to save time in the exam hall!