Question

1. (4 points) 1200 students participated in an exam, and we know that the mean of all scores is 120 points, and the standard deviation is 22 points.

a) suppose the frequency distribution of the data (the scores) was symmetric, what does this information tell us about the student scores?
b) suppose the frequency distribution of the data (the scores) was not normally symmetric, what can we say about the student scores?

Explanation:

Part (a)

Step1: Recall Symmetric Distribution Properties

In a symmetric distribution (e.g., normal, uniform), mean, median, mode are equal or close. The standard deviation ($\sigma = 22$) and mean ($\mu = 120$) help describe spread and center.

Step2: Interpret Symmetric Data

Symmetry implies data is balanced around the mean. Most scores cluster near 120, with 22 as the average deviation. The distribution is likely not skewed, so median ≈ mean, and data is evenly spread on both sides of 120.

Step1: Recall Asymmetric (Skewed) Distribution

In a non - symmetric (skewed) distribution, mean, median, mode differ. The mean (120) and standard deviation (22) still describe center and spread, but skew affects interpretation.

Step2: Interpret Non - Symmetric Data

Without symmetry, the mean may be pulled by skewness (e.g., right - skew: mean > median; left - skew: mean < median). The standard deviation still measures spread, but the distribution has a longer tail on one side. We know the center (mean = 120) and spread (σ = 22), but can't assume balance around the mean (e.g., more scores on one side of 120).

Answer:

In a symmetric distribution, the mean (120) is a good measure of center (median ≈ mean). The standard deviation (22) shows the average score deviation from 120. Data is balanced around 120, with no skew, so most scores are near 120, and spread is symmetric (e.g., similar number of scores above/below 120 ± 22).

Part (b)