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ANSWER TWO QUESTIONS IN THE IMAGE PROVIDED

Let's solve both questions from the image.

### Question 20: Standard Deviation of the Sample
The 7 scores in the sample are: \( 82, 76, 80, 90, 78, 84, 86 \).

We are asked to find the standard deviation using the formula:

\[
s = \sqrt{\frac{N \sum X^2 - (\sum X)^2}{N(N-1)}}
\]

Let's calculate it step by step.

- Step 1: Compute \( \sum X \) (the sum of the scores).
- Step 2: Compute \( \sum X^2 \) (the sum of the squares of the scores).
- Step 3: Substitute these values into the formula and calculate the standard deviation.

Let me calculate that.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, you can manually calculate the standard deviation using the steps outlined:

1. **Sum of scores (\( \Sigma X \)):** Add up all the values.
2. **Sum of squared scores (\( \Sigma X^2 \)):** Square each value and sum them.
3. Apply the formula:
\[
s = \sqrt{\frac{N \Sigma X^2 - (\Sigma X)^2}{N(N-1)}}
\]
Where \(N\) is the number of values.

### Question 21: Shape of the Frequency Distribution

The table shows values of \( X \) and their frequencies \( f \):

\[
\begin{array}{c|c}
X & f \\
\hline
10 & 14 \\
9 & 16 \\
8 & 29 \\
7 & 52 \\
6 & 54 \\
\end{array}
\]

The shape of this distribution is **negatively skewed** (left-skewed), because the frequency increases as \( X \) decreases, which means the tail is on the left side of the distribution.

Let me know if you'd like further details or explanations. Here are 5 related questions for you:

1. How would you calculate variance for this sample?
2. What would the mean of the sample be, given the scores?
3. Can you determine the mode of the frequency distribution?
4. How would a positively skewed distribution look compared to a negatively skewed one?
5. What are the implications of standard deviation in interpreting data variability?

**Tip:** When dealing with distributions, always check how the data points spread around the mean for insights into skewness and variability.

20. If the 7 scores in a sample were 82, 76, 80, 90, 78, 84, and 86, what is the standard deviation? 21. What is the shape of the frequency distribution shown in the table?

Learn how to calculate the standard deviation of a sample dataset and identify the shape of a frequency distribution. This guide covers the statistical concepts of standard deviation and skewness, providing a step-by-step approach for students in Grades 10-12.

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