Question

If two sets of scores have the same mean, then a. they must have the same variability. b. they must have similar variabilities. c. they must have different variabilities. d. their variabilities could be the same or they could be different

Short answer

d. their variabilities could be the same or they could be different.

Step-by-step solution

Understanding the Relationship Between Mean and Variability

The mean is the average value of a data set, while variability measures how spread out the scores are around the mean. Variability includes measures such as range, variance, and standard deviation.

Exploring the Possibilities

While the mean provides information about the central tendency of the data, it does not inherently influence the variability. Sets with the same mean can have different spreads of data.

Considering Variability in Sets With Same Means

Two sets can have identical average (mean) values, yet exhibit different levels of dispersion around that mean. This means they might have similar variabilities, different variabilities, or even the same variabilities.

Conclusion

Given that sets with the same mean can have the same, similar, or completely different variabilities, the correct conclusion is that the variabilities could be the same or they could be different.

Key concepts

Central Tendency

The concept of central tendency refers to the typical value in a data set, also known as its center. It's crucial in statistics because it allows us to understand the average performance or typical case in a dataset. The most common measures of central tendency are the mean, median, and mode.

• Mean is the arithmetic average, calculated by summing all the data points and dividing by the number of data points.
• Median is the middle value when all the numbers are arranged in order.
• Mode is the most frequently occurring value in the dataset.

In the given exercise, mean is used to compare two data sets, suggesting a central reference point for understanding and comparing data. However, it's important to note that while mean gives an idea of the center, it does not provide insights into the variability of the data.

Data Dispersion

Data dispersion describes how spread out the data values are around the central tendency. It's important because two different data sets can have the same central tendency but different dispersion.
There are several measures to understand data dispersion:

• Range: The difference between the highest and lowest data points.
• Interquartile Range (IQR): The range of the middle 50% of the data points, after dividing the data into quartiles.
• Variance and Standard Deviation: These are more complex measures, commonly used for their precise statistical properties.

Dispersion provides insights into the reliability and consistency of the central tendency. In our exercise, even if the mean is the same, dispersion can classify the data sets by how tightly or loosely the scores are spread.

Standard Deviation

Standard deviation is a fundamental measure of data dispersion. It quantifies the amount of variation or spread in a set of data values. A low standard deviation means that the data points tend to be close to the mean, while a high standard deviation indicates data points are spread out over a wider range of values.
The formula to find the standard deviation \[\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \]

• \(x_i\) represents each data point.
• \(\mu\) is the mean of the data.
• \(N\) is the number of data points.

In our problem, even when two sets share the same mean, the standard deviations can be different, indicating different levels of variability.

Variance

Variance is closely related to standard deviation and is another measure of data dispersion. It represents the average of the squared differences from the Mean. Variance helps in understanding how far each number in the data set is from the mean and therefore from every other number in the set.
The formula is:\[\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}\]

• The term \((x_i - \mu)^2\) introduces more weight to data points further from the mean.
• Variance allows for a comprehensive view of data spread compared to mean.

The task showed that multiple data sets might share their mean, yet have different variances, resulting in different interpretations of the data spread.