Question

worksheet 2: center & spread (s.id.2)
part a: distribution shape
the test scores below are shown:
62, 65, 70, 72, 75, 78, 80, 85, 92

1. describe the shape of the distribution (symmetric, skewed left, skewed right).
2. calculate the mean and median.
3. which measure of center best represents the data? explain.

part b: outliers
another class has the following scores:
60, 72, 75, 78, 80, 82, 85, 88, 100

1. identify any outliers.
2. should the mean or median be used? justify your answer.

Explanation:

Step1: Analyze distribution shape

Compare low and high score spreads: Low scores (62,65,70,72,75) have smaller gaps; high scores (78,80,85,92) have larger gaps, pulling the right tail longer.

Step2: Calculate mean of first dataset

Sum all scores, divide by count.
$\text{Mean} = \frac{62+65+70+72+75+78+80+85+92}{9} = \frac{679}{9} \approx 75.44$

Step3: Calculate median of first dataset

Find middle value of ordered data (9 values, 5th is median).
$\text{Median} = 75$

Step4: Evaluate best center measure

Distribution is skewed right; median resists skew better.

Step5: Find quartiles for second dataset

Ordered data: 60,72,75,78,80,82,85,88,100. $Q_1=75$, $Q_3=85$.

Step6: Calculate IQR and outlier bounds

$\text{IQR}=Q_3-Q_1=85-75=10$. Lower bound: $Q_1-1.5\times\text{IQR}=75-15=60$. Upper bound: $Q_3+1.5\times\text{IQR}=85+15=100$.

Step7: Evaluate center measure for outliers

Dataset has no outliers, but check skew; mean is suitable if symmetric.

Answer:

1. Skewed right
2. Mean: $\approx 75.44$, Median: $75$
3. The median best represents the data. The distribution is skewed right, and the median is not affected by the larger spread of high scores (skew) as much as the mean.
4. There are no outliers. All scores fall within the bounds $60 \leq x \leq 100$.
5. The mean can be used. There are no outliers, and the distribution is roughly symmetric, so the mean accurately reflects the center of the data.