Question

1. (l4) the table shows the length of a class period for each of the schools listed.

a. if a new school, blue road high, with a class period of length 110 minutes, is added to the data, determine what effect it would have on the following:
i. standard deviation:
ii. range:
b. with blue road high’s data included, which measure of center would be most affected by the 110 minutes? explain your answer in the context of the problem.
school|length of class (min)
---|---
lakeview high|50
center high|70
oak hill high|75
fairside high|60
jeffries high|65
rodgers high|65
new hill high|60
sunnyville high|55
pine hill high|65
greenville high|70
faith high|60

Explanation:

Step1: Understand standard - deviation concept

Standard deviation measures the spread of data. A new extreme value will increase spread.

Step2: Analyze the data

The original data has class - period lengths mostly in the 50 - 75 range. 110 is an outlier. Adding it will increase the differences from the mean, thus increasing the standard deviation.

Step3: Understand range concept

Range is the difference between the maximum and minimum values in a data - set.

Step4: Determine the original range

Original maximum is 75, minimum is 50, so original range is \(75 - 50=25\).

Step5: Determine the new range

With 110 added, new maximum is 110, minimum is 50, new range is \(110 - 50 = 60\). So the range increases.

Step6: Understand measures of center

The main measures of center are mean, median. The median is the middle value when data is ordered, and the mean is the sum of all values divided by the number of values.

Step7: Analyze the effect on median

Original data has 10 values. Median is the average of the 5th and 6th ordered values. With 11 values (after adding Blue Road High), the median is the 6th ordered value. The outlier 110 may not change the position of the middle value much.

Step8: Analyze the effect on mean

The mean is calculated as \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). The outlier 110 will increase the sum \(\sum_{i = 1}^{n}x_{i}\) and thus have a large impact on the mean.

Answer:

a. i. The standard deviation will increase.
ii. The range will increase.
b. The mean will be most affected. The mean is calculated by summing all data values and dividing by the number of data points. An outlier like 110 will significantly increase the sum and thus change the mean. The median is less affected as it is based on the position of values in an ordered list and the outlier may not change the middle - position value as much.