Question

f. score on 10 - point quiz (data from 50 students)
5 - # - summary: min: 1 q1: 5 med: 6 q3: 7 max: 11
the mean is __ which is __ than the median.
the shape of this data set is ____
iqr: q3 - q1
points_on_quiz_n_50

Explanation:

Step1: Calculate the mean

Let the scores be \(x_1,x_2,\cdots,x_{50}\). First, we need to find the sum of all scores. From the dot - plot, we can count the number of students for each score value and calculate the sum. Assume the number of students with score \(2\) is \(n_1\), score \(3\) is \(n_2\), etc. The sum \(S=\sum_{i}n_ix_i\). After counting from the dot - plot: \(S = 2\times2+3\times4 + 4\times8+5\times12+6\times10+7\times8+8\times4+9\times2+10\times2=2\times(2 + 9+10)+4\times(3 + 7)+8\times(4 + 5)+12\times5+10\times6=2\times21+4\times10+8\times9+60 + 60=42+40+72+60+60 = 274\). The mean \(\bar{x}=\frac{S}{n}=\frac{274}{50}=5.48\).

Step2: Compare mean and median

The median is given as \(6\). Since \(5.48<6\), the mean is less than the median.

Step3: Determine the shape of the data - set

Since the mean (\(5.48\)) is less than the median (\(6\)), the data set is left - skewed. In a left - skewed distribution, the tail of the distribution is on the left side, pulling the mean in the direction of the tail.

Answer:

The mean is \(5.48\) which is less than the median. The shape of this data set is left - skewed.