To find the number of data points within 2 population standard deviations of the mean, we will follow these steps:

Step 1: Calculate the mean

We use the formula for the mean of a frequency distribution:

$\bar{x} = \frac{\sum (x_i \cdot f_i)}{\sum f_i}$

Where:

• $x_i$ is the data point
• $f_i$ is the frequency of each data point

$\bar{x} = \frac{(10 \cdot 2) + (18 \cdot 8) + (22 \cdot 10) + (23 \cdot 16) + (26 \cdot 16) + (31 \cdot 14) + (34 \cdot 10) + (36 \cdot 7) + (40 \cdot 2)}{2 + 8 + 10 + 16 + 16 + 14 + 10 + 7 + 2}$

Let's compute this.

Step 2: Calculate the population standard deviation

The formula for the population standard deviation is:

$\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}}$

Once we have the mean, we can compute the standard deviation.

Step 3: Find the range within 2 standard deviations

After finding the standard deviation $\sigma$, we compute the range of data within 2 standard deviations:

$\bar{x} - 2\sigma \leq x \leq \bar{x} + 2\sigma$

Step 4: Count the number of data points

Finally, we'll count how many data points fall within this range based on their frequencies.

Let me calculate this for you.### Results:

• Mean ($\bar{x}$): 26.75
• Standard deviation ($\sigma$): 6.29
• Range within 2 standard deviations: [14.18, 39.33]

There are 81 data points (based on the given frequencies) that fall within 2 population standard deviations of the mean.

Would you like more details, or do you have any questions?

5 Related Questions:

1. How is the population standard deviation different from the sample standard deviation?
2. How would the results change if the frequencies of the data were altered?
3. What is the significance of using 2 standard deviations in this type of analysis?
4. How do frequency distributions affect the mean and standard deviation?
5. How can this method be extended to different types of data distributions?

Tip:

Understanding the spread of data within standard deviations helps you assess how clustered or spread out your data is from the mean.