Question

each of 6 students reported the number of movies they saw in the past year. this is what they reported.
15, 12, 9, 7, 20, 8
find the median and mean number of movies that the students saw.
if necessary, round your answers to the nearest tenth.
(a) median: movies
(b) mean: movies

Explanation:

Part (a): Median Calculation

Step1: Order the data

First, we need to order the data set from smallest to largest. The given data is \(15, 12, 9, 7, 20, 8\). When we order it, we get \(7, 8, 9, 12, 15, 20\).

Step2: Find the median position

For a data set with \(n\) values, if \(n\) is even, the median is the average of the \(\frac{n}{2}\)-th and \((\frac{n}{2}+1)\)-th values. Here, \(n = 6\) (which is even), so \(\frac{n}{2}=\frac{6}{2}=3\) and \(\frac{n}{2}+1 = 4\).

Step3: Calculate the median

The 3rd value in the ordered data set is \(9\) and the 4th value is \(12\). The median is the average of these two values, so we calculate \(\frac{9 + 12}{2}\).
\[
\frac{9+12}{2}=\frac{21}{2}=10.5
\]

Part (b): Mean Calculation

Step1: Sum the data values

To find the mean, we first sum all the data values. The data values are \(15, 12, 9, 7, 20, 8\). The sum \(S\) is calculated as:
\[
S=15 + 12+9 + 7+20 + 8
\]
\[
S=(15 + 12)+(9 + 7)+(20 + 8)=27+16 + 28=71
\]

Step2: Calculate the mean

The mean \(\bar{x}\) is the sum of the data values divided by the number of data values \(n\). Here, \(n = 6\), so:
\[
\bar{x}=\frac{S}{n}=\frac{71}{6}\approx11.8
\]

Answer:

(a) Median: \(10.5\) movies
(b) Mean: \(11.8\) movies