Question

If a set of 20 different numbers has its smallest and largest values removed, how will that affect the standard deviation of the set? (A) The standard deviation will increase. (B) The standard deviation will decrease. (C) The standard deviation will remain the same. (D) Not enough information is provided.

Short answer

(B) The standard deviation will decrease.

Step-by-step solution

Understand Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells us how much the numbers in a set deviate from the mean (average) of that set.

Impact of Removing Extremes

When the smallest and largest values in a set are removed, the range of the data set decreases. This typically reduces the deviations of the remaining numbers from the mean, because there are no extreme values to create larger deviations.

Predict the Change in Standard Deviation

Removing the extreme values generally causes the standard deviation to decrease. This is because the data becomes more concentrated around the mean, thereby showing less overall spread or dispersion.

Consider Alternative Effects

In rare cases, if the data distribution is very unusual, removing the extremes may have a different effect. However, without additional specific information about the distribution of the numbers, the most likely effect is a reduction in standard deviation.

Key concepts

Data Set

A "data set" is simply a collection of numbers. Each number in a data set is referred to as a "data point." Data sets can be found everywhere, from the list of your monthly expenses to the scores of your favorite sports team over a season.
The numbers in a data set can be used to perform statistical analysis, which helps us understand patterns and trends. In the exercise, the data set consists of 20 different numbers. This means that there are 20 distinct numerical values being considered for analysis. By understanding the structure of a data set, we can begin to explore aspects like variation and mean.

Variation

"Variation" is a term used to describe how spread out or dispersed a set of numbers is. Understanding the variation in a data set is crucial because it gives insight into its consistency.
In the context of the exercise, the variation is related to the standard deviation. When we talk about removing the smallest and largest numbers from a set, we are directly affecting the variation. With fewer extreme values, the remaining data tends to be more clustered around the mean, indicating lower variation.

Mean

The "mean," also known as the average, is one of the most fundamental concepts in statistics. To find the mean of a data set, you sum up all the numbers and then divide by the number of data points. In the context of the given exercise, the mean helps us understand the central tendency of the data before and after removing extreme values.
When extreme values (the lowest and highest numbers) are removed, the mean of the remaining data can shift slightly, but usually, the concentration of remaining numbers around the new mean decreases the variation, affecting the standard deviation.