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, 10, 10, 10, 15,15,15,19, 20, 20, 20, 25, 25, 25, 30, 30, 55, 55

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 8 above 10 to 19, up to 6 above 20 to 29, up to 2 above 30 to 39, and up to 2 above 50 to 59. There is no shaded bar above 40 to 49.

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

The IQR of 10 is the most accurate to use, since the data is skewed.
The range of 10 is the most accurate to use, since the data is skewed.
The IQR of 45 is the most accurate to use to show that they need more money.
The range of 45 is the most accurate to use to show that they have plenty of money.

Answer 1

The answer would be 10

Answer 2

In this case, the data is skewed to the right, with a long tail of higher values. This means that measures of central tendency such as the mean or median may not accurately represent the "typical" donation amount, as they can be heavily influenced by the higher values.

Therefore, a measure of variability that takes into account the spread of the data and is less influenced by outliers would be more appropriate. The interquartile range (IQR) is a measure of variability that is resistant to outliers and is defined as the difference between the third quartile (Q3) and the first quartile (Q1). It tells us the spread of the middle 50% of the data.

In this case, the IQR would be calculated as follows:

Q1 = 15 (median of the lower half of the data)

Q3 = 25 (median of the upper half of the data)

IQR = Q3 - Q1 = 25 - 15 = 10

Therefore, the charity should use the IQR of 10 to accurately represent the spread of the data. This tells us that the middle 50% of the donations are between $15 and $25.