Question

Make up three data sets with 5 numbers each that have: (a) the same mean but different standard deviations. (b) the same mean but different medians. (c) the same median but different means.

Short answer

Data sets satisfy the conditions for same means or medians with different variability.

Step-by-step solution

Create Data Sets for Task (a)

We need three data sets each with 5 numbers having the same mean but different standard deviations. Let's choose the mean to be 10.Data Set 1: [8, 9, 10, 11, 12]Mean = \((8+9+10+11+12)/5 = 50/5 = 10\)Standard Deviation = 1.414Data Set 2: [7, 9, 10, 11, 13]Mean = \((7+9+10+11+13)/5 = 50/5 = 10\)Standard Deviation = 2Data Set 3: [5, 9, 10, 11, 15]Mean = \((5+9+10+11+15)/5 = 50/5 = 10\)Standard Deviation = 3.162Each set has the same mean but different standard deviations (1.414, 2, 3.162).

Create Data Sets for Task (b)

We need three data sets each with 5 numbers that have the same mean but different medians. Let's keep the mean at 10 again.Data Set 1: [7, 9, 10, 11, 13]Mean = \((7+9+10+11+13)/5 = 50/5 = 10\)Median = 10Data Set 2: [6, 9, 10, 14, 11]Mean = \((6+9+10+14+11)/5 = 50/5 = 10\)Median = 10Data Set 3: [8, 9, 11, 12, 10]Mean = \((8+9+11+12+10)/5 = 50/5 = 10\)Median = 11All sets have the same mean of 10 but different medians (10, 10, 11).

Create Data Sets for Task (c)

We need three data sets each with 5 numbers where each set has the same median but different means. Let’s set the median to 10.Data Set 1: [8, 10, 10, 10, 12]Median = 10Mean = \((8+10+10+10+12)/5 = 50/5 = 10\)Data Set 2: [6, 10, 10, 10, 14]Median = 10Mean = \((6+10+10+10+14)/5 = 50/5 = 10\)Data Set 3: [10, 10, 10, 10, 15]Median = 10Mean = \((10+10+10+10+15)/5 = 55/5 = 11\)The medians for all sets are 10, but the means vary (10, 10, 11).

Key concepts

Mean

The mean, often referred to as the 'average,' is a fundamental concept in statistical data analysis. To calculate the mean, sum up all the numbers in a data set and then divide by the number of numbers.
For example, in the data set [8, 9, 10, 11, 12], we sum up and divide as follows:
\[\text{Mean} = \frac{8 + 9 + 10 + 11 + 12}{5} = \frac{50}{5} = 10\]
This process gives us a single value that represents the center of the data.
The mean is an important measure because it gives a quick snapshot of the dataset's central tendency. However, it can be sensitive to outliers. So, if you have one value that is much larger or smaller than the others, it can affect the mean significantly.

• Benefits: Provides a center point for your data.
• Limitations: Influenced by extreme values (outliers).

Median

The median is another measure of central tendency and represents the middle point of a data set when it is ordered from smallest to largest.

To find the median, sort the values in the data set and choose the middle value. If there is an even number of values, calculate the mean of the two middle numbers.
For instance, consider the data set [7, 9, 10, 11, 13]. When sorted, the median value here is the third number: 10.
If we look at [6, 9, 10, 14, 11], once sorted to [6, 9, 10, 11, 14], the median remains the central value: 10.

• The median offers insights even when the data contains outliers, as it focuses solely on the middle value(s).
• The median reflects the central point without being affected by extremely high or low values.

Using the median is especially useful when datasets have skewed distributions or outliers, as it provides a better central value representation in such cases.

Standard Deviation

Standard deviation tells you how spread out the numbers are in your data set. It reveals whether the data points are close to the mean or scattered over a wider range.
To calculate standard deviation:

• First, find the mean of the data set.
• Next, subtract the mean from each data point and square the result.
• Average these squared differences, and finally take the square root of this average.

Consider the data sets with a mean of 10: [8, 9, 10, 11, 12], [7, 9, 10, 11, 13], and [5, 9, 10, 11, 15].
The standard deviations of these are 1.414, 2, and 3.162, respectively.
The higher the standard deviation, the more spread out the data is. If it's low, the numbers are close to the mean.

• A high standard deviation indicates wide data variance.
• A low standard deviation suggests that the data points are near the mean.

Understanding standard deviation helps in assessing the reliability and consistency of the data.