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

Step 1: Calculate the mean

The mean (μ) is calculated by finding the weighted average of the data points, using their frequencies.

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

Where:

• $x_i$ is the data value.
• $f_i$ is the frequency of the data value.

Step 2: Calculate the population standard deviation

The population standard deviation ($\sigma$) is calculated using the formula:

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

Step 3: Determine the interval

Once we have $\mu$ and $\sigma$, we calculate the interval $[\mu - 2\sigma, \mu + 2\sigma]$, which represents the range within 2 standard deviations of the mean.

Step 4: Count the number of data points within this range

Finally, we sum the frequencies of all data points that lie within this interval.

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Let's proceed with the calculation:

Data:

Data (x)	Frequency (f)
0	1
1	6
3	11
4	16
5	20
6	16
9	9
14	8
15	2

Now I'll calculate the mean, standard deviation, and the number of data points within 2 standard deviations.### Results:

• Mean (μ): 5.87 (approximately)
• Population Standard Deviation (σ): 3.55 (approximately)
• Interval within 2 Standard Deviations: [-1.24, 12.97]

The data points within this interval are from 0 to 9. The total number of data points that lie within this range is 79.

Would you like further details or have any questions?

Here are 5 related questions to extend this topic:

1. How would the result change if we used sample standard deviation instead of population standard deviation?
2. How can the frequency distribution affect the variability of the data?
3. What is the probability of a data point falling within 1 standard deviation of the mean?
4. How does the shape of the data distribution (e.g., skewed or symmetric) affect the mean and standard deviation?
5. What if there were more extreme outliers in the data? How would that impact the calculation?

Tip: In a normal distribution, about 95% of the data falls within 2 standard deviations from the mean.