Chapter 3, Problem 1CQ

Of the mean, median, and mode, which must be a value that actually appears in the data set?

To determine

Find the measure which must be a value in the data set.

Answer to Problem 1CQ

The mode must appear in the data set.

Explanation of Solution

Mean:

The arithmetic mean (also called the mean) is the most commonly used measure of central tendency. It is calculated by summing the observed numerical values of a variable in a set of data and then dividing the total by the number of observations involved.

Computation of mean:

If $x_1,x_2\mathrm{...}{x}_n$ be n values.

The mean value is calculated as,

$\begin{array}{c}\text{Mean}=\frac{\text{sum of all values}}{\text{number of values}}\\ =\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\end{array}$

Therefore, mean is the average of all values. Sometimes the data set contains the mean but it is not possible that always mean value will be an observation in the data.

Median:

The median is the middle most value of the observation. In case of no tie, half of the observations will be less than middle most value, and half of the observations will be greater than that value. In addition, the median is unaffected by extreme values in a set of data.

Let $x_1,x_2\mathrm{...}{x}_n$ be n values.

The steps for finding the median:

• The all data values should be arranged in ascending order.
• If the total number of data values, n is odd, then the median will be the middle value or if n is even, then the median will be the average of middle two values.

Hence, it can be said that the median is actually the middle value that divides the whole data set in two parts. For an even sample size, the median will be average. Hence, median does not appear in the data.

Mode:

Occasionally, when describing a set of data, the mode is used as a measure of central tendency. The mode is the most frequently occurring value in a data set. The mode can often be identified by referring to a histogram or polygon. Distributions with a single mode are referred to as unimodal. Distributions with two modes are referred to as bimodal.

Therefore, mode is the data value with maximum frequency. Hence, mode appears in the data.

• Hence, it can be said that mode must appear in the data set compared to mean, median and mode.