Question

A charity needs to report its typical donations received. The following is a list of the donations from one week. A histogram is provided to display the data.

10, 11, 35, 39, 40, 42, 42, 45, 49, 49, 51, 51, 52, 53, 53, 54, 56, 59

A graph titled Donations to Charity in Dollars. The x-axis is labeled 10 to 19, 20 to 29, 30 to 39, 40 to 49, and 50 to 59. The y-axis is labeled Frequency. There is a shaded bar up to 2 above 10 to 19, up to 2 above 30 to 39, up to 6 above 40 to 49, and up to 8 above 50 to 59. There is no shaded bar above 20 to 29.

Which measure of variability should the charity use to accurately represent the data? Explain your answer.

The range of 13 is the most accurate to use, since the data is skewed.
The IQR of 49 is the most accurate to use to show that they need more money.
The range of 49 is the most accurate to use to show that they have plenty of money.
The IQR of 13 is the most accurate to use, since the data is skewed.

Answer 1

The IQR (interquartile range) of 13 is the most accurate measure of variability to use in this case, since the data is skewed and contains some outliers.

The IQR is a measure of variability that is less sensitive to extreme values than the range, and is calculated as the difference between the upper and lower quartiles (the 75th and 25th percentiles). It provides a measure of the spread of the middle 50% of the data, which is useful for understanding the typical range of donations received.

In this case, the IQR is calculated as follows:

- The median of the data is 51 (the value in the middle).

- The lower quartile (Q1) is the median of the lower half of the data, which is 42.

- The upper quartile (Q3) is the median of the upper half of the data, which is 54.

- The IQR is the difference between Q3 and Q1: IQR = Q3 - Q1 = 54 - 42 = 12.

So the IQR of 13 is a useful measure of variability to use for this data set, since it captures the spread of the middle 50% of the data while being less sensitive to the outliers at the higher end of the distribution.