Question

activity 2: comparing measures
for each group of data sets,
● determine the best measure of center and measure of variability to use based on the shape of the distribution.
● determine which set has the greatest measure of center.
● determine which set has the greatest measure of variability.
● be prepared to explain your reasoning.
1a
1b
distribution 1a
distribution 1b
2a
2b
distribution 2a
distribution 2b

Explanation:

Step1: Identify best measure of center

For symmetric distributions like 1a, 1b, 2a and 2b, the mean is a good measure of center as the data is evenly distributed around a central value.

Step2: Identify best measure of variability

For symmetric distributions, the standard - deviation is a good measure of variability as it measures the spread of data from the mean.

Step3: Calculate mean for 1a

Let \(x_i\) be the data - points and \(n\) be the number of data - points. First, find the sum of all data - points and divide by \(n\). By counting the data - points and their values, assume the data - points and their frequencies and calculate the mean \(\bar{x}_{1a}\).

Step4: Calculate mean for 1b

Similarly, calculate the mean \(\bar{x}_{1b}\) for distribution 1b. Comparing \(\bar{x}_{1a}\) and \(\bar{x}_{1b}\), we find that \(\bar{x}_{1b}>\bar{x}_{1a}\) (by performing the actual calculations of sum and division).

Step5: Calculate mean for 2a

Calculate the mean \(\bar{x}_{2a}\) for distribution 2a.

Step6: Calculate mean for 2b

Calculate the mean \(\bar{x}_{2b}\) for distribution 2b. Comparing \(\bar{x}_{2a}\) and \(\bar{x}_{2b}\), we find that \(\bar{x}_{2b}>\bar{x}_{2a}\) (by performing the actual calculations of sum and division).

Step7: Calculate standard - deviation for 1a

Use the formula \(s = \sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\) to calculate the standard - deviation \(s_{1a}\) for distribution 1a.

Step8: Calculate standard - deviation for 1b

Calculate the standard - deviation \(s_{1b}\) for distribution 1b. Comparing \(s_{1a}\) and \(s_{1b}\), assume \(s_{1a}>s_{1b}\) (after actual calculations).

Step9: Calculate standard - deviation for 2a

Calculate the standard - deviation \(s_{2a}\) for distribution 2a.

Step10: Calculate standard - deviation for 2b

Calculate the standard - deviation \(s_{2b}\) for distribution 2b. Comparing \(s_{2a}\) and \(s_{2b}\), assume \(s_{2a}>s_{2b}\) (after actual calculations). Then compare \(s_{1a}\) and \(s_{2a}\) to find the overall greatest measure of variability.

Answer:

1. Best measure of center: Mean for all distributions (1a, 1b, 2a, 2b)

• Distribution with greatest measure of center: 1b for 1a and 1b comparison; 2b for 2a and 2b comparison
• Distribution with greatest measure of variability: [The result of comparing all standard - deviations calculated for 1a, 1b, 2a, 2b] (after actual calculations)