To find the standard deviation for the given data, we first extract the actual data values from the stem-and-leaf plot. Here’s how we interpret the data:

Data interpretation from the stem-and-leaf plot:

• Stems represent the tens place.
• Leaves represent the units place.

Step 1: List the Data Values

Based on the stem and leaf structure provided:

• $1 | 6$ means 16
• $2 | 2$ and $2 | 6$ mean 22 and 26
• $3 | 2$ and $3 | 6$ mean 32 and 36

So, the dataset is: $16, 22, 26, 32, 36$

Step 2: Calculate the Mean (Average)

The mean $\mu$ is calculated as: $\mu = \frac{16 + 22 + 26 + 32 + 36}{5} = \frac{132}{5} = 26.4$

Step 3: Calculate Each Value's Deviation from the Mean, Square It, and Find the Average

For each value $x$, calculate $(x - \mu)^2$:

1. For $x = 16$: $(16 - 26.4)^2 = (-10.4)^2 = 108.16$
2. For $x = 22$: $(22 - 26.4)^2 = (-4.4)^2 = 19.36$
3. For $x = 26$: $(26 - 26.4)^2 = (-0.4)^2 = 0.16$
4. For $x = 32$: $(32 - 26.4)^2 = (5.6)^2 = 31.36$
5. For $x = 36$: $(36 - 26.4)^2 = (9.6)^2 = 92.16$

Sum of squared deviations: $108.16 + 19.36 + 0.16 + 31.36 + 92.16 = 251.2$

Step 4: Find the Variance and Standard Deviation

The variance $\sigma^2$ is the average of these squared deviations: $\sigma^2 = \frac{251.2}{5} = 50.24$

The standard deviation $\sigma$ is the square root of the variance: $\sigma = \sqrt{50.24} \approx 7.09$

Answer

The standard deviation is approximately 7.09.

Would you like further clarification on any step?

Here are some related questions you might find helpful:

1. How is the variance related to standard deviation?
2. What is the difference between population and sample standard deviation?
3. Why do we square the deviations when calculating variance?
4. How does standard deviation interpret the spread of data?
5. How would the standard deviation change if more values were added?

Tip: Always double-check your data extraction from stem-and-leaf plots to ensure accurate calculations.