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Chapter 4: Problem 24

Give two sets of five numbers that have the same mean but different standard deviations, and give two sets of five numbers that have the same standard deviation but different means.

Short Answer

Expert verified
Two sets of five numbers with the same mean but different standard deviations are [3, 3, 3, 3, 3] and [1, 2, 3, 4, 5]. Two sets of five numbers with the same standard deviation but different means are [2, 3, 4, 5, 6] and [12, 13, 14, 15, 16].

Step by step solution

01

Define Sets with Same Mean and Different Standard Deviations

To achieve this, one should pick a common mean and then distribute the numbers around that mean differently for each set. For example, let's take the mean as 3. The first set could be [3, 3, 3, 3, 3], and the second set could be [1, 2, 3, 4, 5]. Both have the same mean but different standard deviations since the numbers in the second set are more spread out.
02

Calculate the Standard Deviations

By calculation, the standard deviation for the first set (using the formula for standard deviation in a sample) is 0 since all the values are equal, and the standard deviation for the second set is approximately 1.58.
03

Define Sets with Same Standard Deviation and Different Means

To get this, one can first pick a common standard deviation and then create sets where the numbers are as spread out from different means. For example, we could keep the standard deviation as 1.58 and then the numbers could be as spread out from the mean. The first set could be [2, 3, 4, 5, 6] and the second set [12, 13, 14, 15, 16]. The numbers are as spread from the mean, but the central value is different.
04

Calculate the Means

By calculation, the mean for the first set is 4 and for the second set is 14. Hence, they have different means but the same standard deviation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean
The mean, often referred to as the average, is a central measure in statistics. To calculate the mean of a data set, sum up all the values and then divide by the total number of values. This gives a central value that represents the typical amount in your set.
For instance, in the data set [1, 2, 3, 4, 5], the mean is calculated as follows:
  • Add up all the numbers: 1 + 2 + 3 + 4 + 5 = 15
  • Divide by the total count of numbers: 15 divided by 5 equals 3
So, the mean of this data set is 3, offering a simple glimpse into where values tend to cluster.
The mean, however, doesn't express the spread of the data, which is where the standard deviation comes into play.
Standard Deviation
Standard deviation is a measure of how spread out the numbers in a data set are from the mean. It provides insight into the variability present in your data.
Let鈥檚 consider two sets with the same mean but different spreads: [3, 3, 3, 3, 3] and [1, 2, 3, 4, 5].
  • For the first set, since all numbers are 3, the numbers do not deviate from the mean, resulting in a standard deviation of 0.
  • In the second set, numbers vary around the mean, leading to a larger standard deviation.
This illustrates how standard deviation can stay the same across different data sets with similar spreads despite contrasting means.
Data Sets
Data sets are collections of numbers or values that you analyze statistically. They can reveal trends and patterns to better understand the data.
For example, to create two data sets with the same standard deviation but different means, consider these sets:
  • [2, 3, 4, 5, 6]
  • [12, 13, 14, 15, 16]
Although both sets have a different central value, they share a similar spread. This helps us see how different data sets can maintain the same measure of variability even while the central tendency shifts.
Variability
Variability describes how data points differ from each other and from the mean. It's crucial for understanding the reliability and predictability of your data.
In our exercise, the key was to observe data sets with differing levels and patterns of variability. Variability can give us a better idea of data consistency. A low variability means data points are close to the mean, as in [3, 3, 3, 3, 3].
On the other hand, a higher variability in a set like [1, 2, 3, 4, 5] indicates more diversity in data points, impacting decisions and predictions made from the data.
Understanding variability alongside the mean and standard deviation is essential in the field of descriptive statistics.

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