Question

A teacher is looking at the set of recent test scores from her class. After investigating, she realized that she transposed two different test grades. The 19 should be 91 , while another student who had an 86 should have had a 68. After correcting these two mistakes, which of the following must increase in the updated data set? A) The range B) The standard deviation C) The median D) The mean

Short answer

The correct answer is D) The mean, since there is an overall increase in the dataset and the number of values doesn't change.

Step-by-step solution

Understand the data change

We are given that a 19 was changed to a 91 and an 86 was changed to a 68. The change in values can be summarized as: - 19 -> 91 (increase of 72) - 86 -> 68 (decrease of 18) Let's now check each property to see if its value would increase in the updated data set.

The range

The range is the difference between the highest and lowest values in the dataset. In this case, we can see that one of the changed values (19 → 91) is an increase, which could potentially increase the maximum value in the dataset, thus leading to an increased range. So, the range (A) could increase in the updated dataset.

Standard deviation

The standard deviation is the measure of the spread or the dispersion of the values in the dataset. Changing individual data points can affect the spread of the data. In our case, one of the changes (19 → 91) seems to increase the spread of the data, while the other (86 → 68) seems to decrease it. It's not clear from this information whether the standard deviation must increase, so we cannot determine if it's B) the standard deviation.

The median

The median is the middle value of a dataset when arranged in ascending or descending order. Changing one or two values in the dataset might affect the median, but this depends on the specific dataset and the position of the changed values. From the given information, we cannot guarantee that the median (C) must increase in the updated dataset.

The mean

The mean is the sum of all the values in the dataset divided by the number of values. We can analyze the effect of the changes on the mean by calculating the overall change in the dataset. Adding the changes in the values: 72 (increase from 19 to 91) - 18 (decrease from 86 to 68) = 54 (overall increase) Since there is an overall increase in the dataset and the number of values doesn't change, the mean (D) must increase in the updated dataset. The correct answer is D) The mean.