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.

Key concepts

Mean Calculation

Understanding the mean of a data set is crucial for interpreting the overall trend in a given set of numbers. The mean, often known as the average, is calculated by summing up all the data values and then dividing by the number of values. In mathematical terms, if we have a data set represented as: \( x_1, x_2, ..., x_n \), the mean \( \bar{x} \) is given by:\[\bar{x} = \frac{x_1 + x_2 + ... + x_n}{n}\]Let's apply this with a practical perspective using the example from the exercise:

• Original Change: 19 to 91, which is an increase of 72.
• Another Change: 86 to 68, which is a decrease of 18.
• Total Change in Values: 72 - 18 = 54, leading to a net increase in data set sum.

The number of total data points remains unchanged. Hence, the mean must reflect this net increase by also increasing. This confirms option D is correct.

Standard Deviation

Standard deviation is a measure that quantifies the amount of variation or dispersion within a set of data values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation means the values are spread out over a larger range. The formula to calculate standard deviation for a population is:\[\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}\]where \( \mu \) is the mean and \( x_i \) represents individual data points.
In the scenario provided:

• Changing 19 to 91 increases the spread of data, potentially raising the standard deviation.
• Conversely, changing 86 to 68 constrains the data range, possibly lowering the deviation.

However, because these changes counterbalance each other to some extent, the overall effect on the standard deviation isn't determined solely by these changes.

Data Range

Data range is a fundamental concept in statistics that provides the simplest measure of data spread. It is calculated as the difference between the maximum and minimum values in the data set:\[\text{Range} = \text{Maximum Value} - \text{Minimum Value}\]Let's see how this applies to our given scenario:

• Original Change: 19 to 91, a significant increase that makes 91 possibly the new maximum value.
• Therefore, since the maximum increases, the data range must also increase.

This adjustment affects the range directly, supporting the idea that it could increase. However, the fundamental minimum value assessment is needed to ensure the complete understanding of range adjustments.

Median in Statistics

The median is the middle value of a data set when it is ordered from lowest to highest. If the data set has an odd number of observations, the median is the value at the \( \left(\frac{n+1}{2}\right) \) position in the ordered list. For an even number of observations, it is the average of the values at the \( \left(\frac{n}{2}\right) \) and \( \left(\frac{n}{2}+1\right) \) positions.
In analyzing how changes can affect the median:

• Adjusting 19 to 91 could position 91 as an upper outlier, significantly altering the arrangement of values.
• The swap from 86 to 68 may impact mid-range values, but assessing exact impact depends on dataset specifics.

Thus, without complete data distribution details, determining whether these changes affect the median is speculative. Changes in positions of individual scores are critical for clear median analysis.