1. For the following exercise, complete the following:

a. Find the mean, median, and range for each of the two data sets.

b. Find the standard deviation using the rule of thumb for each of the data sets.

c. Compare the two sets and describe what you discover.

The following data sets shows the ages of the first seven presidents (President Washington through President Jackson) and the seven most recent presidents including President Obama. Age is given at time of inauguration.

First 7: 57 61 57 57 58 57 61

Second 7: 61 52 69 64 46 54 47

a. To find the mean, median, and range for each data set, we need to follow these steps:

For the first data set (first 7 presidents: 57, 61, 57, 57, 58, 57, 61):

Mean:
Step 1: Add up all the values: 57 + 61 + 57 + 57 + 58 + 57 + 61 = 408.
Step 2: Divide the sum by the number of values: 408 / 7 = 58.29 (rounded to two decimal places).
So, the mean of the first data set is approximately 58.29.

Median:
Step 1: Arrange the values in ascending order: 57, 57, 57, 57, 58, 61, 61.
Step 2: Identify the middle value(s). Since there are 7 values, the middle value is the (7+1) / 2 = 4th value, which is 57.
So, the median of the first data set is 57.

Range:
Step 1: Find the maximum and minimum values: maximum = 61, minimum = 57.
Step 2: Calculate the difference: 61 - 57 = 4.
So, the range of the first data set is 4.

For the second data set (second 7 presidents: 61, 52, 69, 64, 46, 54, 47):

Mean:
Step 1: Add up all the values: 61 + 52 + 69 + 64 + 46 + 54 + 47 = 393.
Step 2: Divide the sum by the number of values: 393 / 7 = 56.14 (rounded to two decimal places).
So, the mean of the second data set is approximately 56.14.

Median:
Step 1: Arrange the values in ascending order: 46, 47, 52, 54, 61, 64, 69.
Step 2: Identify the middle value(s). Since there are 7 values, the middle value is the (7+1) / 2 = 4th value, which is 54.
So, the median of the second data set is 54.

Range:
Step 1: Find the maximum and minimum values: maximum = 69, minimum = 46.
Step 2: Calculate the difference: 69 - 46 = 23.
So, the range of the second data set is 23.

b. To find the standard deviation using the rule of thumb for each data set, we need to follow these steps:

For the first data set (first 7 presidents: 57, 61, 57, 57, 58, 57, 61):

Step 1: Find the mean: 58.29 (rounded to two decimal places).
Step 2: Find the difference between each value and the mean.
For 57: 57 - 58.29 = -1.29
For 61: 61 - 58.29 = 2.71
For 57: 57 - 58.29 = -1.29
For 57: 57 - 58.29 = -1.29
For 58: 58 - 58.29 = -0.29
For 57: 57 - 58.29 = -1.29
For 61: 61 - 58.29 = 2.71
Step 3: Square each difference: (-1.29)^2, (2.71)^2, (-1.29)^2, (-1.29)^2, (-0.29)^2, (-1.29)^2, (2.71)^2.
Step 4: Calculate the mean of the squared differences: ( (-1.29)^2 + (2.71)^2 + (-1.29)^2 + (-1.29)^2 + (-0.29)^2 + (-1.29)^2 + (2.71)^2 ) / 7 = 2.78 (rounded to two decimal places).
Step 5: Take the square root of the mean obtained in Step 4: sqrt(2.78) = 1.67 (rounded to two decimal places).
So, the standard deviation of the first data set is approximately 1.67.

For the second data set (second 7 presidents: 61, 52, 69, 64, 46, 54, 47):

Step 1: Find the mean: 56.14 (rounded to two decimal places).
Step 2: Find the difference between each value and the mean.
For 61: 61 - 56.14 = 4.86
For 52: 52 - 56.14 = -4.14
For 69: 69 - 56.14 = 12.86
For 64: 64 - 56.14 = 7.86
For 46: 46 - 56.14 = -10.14
For 54: 54 - 56.14 = -2.14
For 47: 47 - 56.14 = -9.14
Step 3: Square each difference: (4.86)^2, (-4.14)^2, (12.86)^2, (7.86)^2, (-10.14)^2, (-2.14)^2, (-9.14)^2.
Step 4: Calculate the mean of the squared differences: ( (4.86)^2 + (-4.14)^2 + (12.86)^2 + (7.86)^2 + (-10.14)^2 + (-2.14)^2 + (-9.14)^2 ) / 7 = 71.61 (rounded to two decimal places).
Step 5: Take the square root of the mean obtained in Step 4: sqrt(71.61) = 8.47 (rounded to two decimal places).
So, the standard deviation of the second data set is approximately 8.47.

c. Comparing the two data sets, we can observe the following:

- The mean of the first data set (58.29) is higher than the mean of the second data set (56.14).
- The median of the first data set (57) is lower than the median of the second data set (54).
- The range of the first data set (4) is smaller than the range of the second data set (23).
- The standard deviation of the first data set (approximately 1.67) is significantly smaller than the standard deviation of the second data set (approximately 8.47).

These differences suggest that the ages of the first 7 presidents were relatively close to each other, while the ages of the most recent 7 presidents had a larger spread.

To solve this exercise, we need to find the mean, median, and range for each data set, calculate the standard deviation using the rule of thumb, and then compare the two sets. Here's a step-by-step guide on how to do that:

a. Finding the Mean, Median, and Range:

1. For the first data set (first 7 presidents):
- Mean: Add up all the numbers (57 + 61 + 57 + 57 + 58 + 57 + 61) and divide by the total count (7). The mean is 58.
- Median: Arrange the numbers in ascending order (57, 57, 57, 57, 58, 61, 61) and find the middle value. Since there are 7 numbers, the median is the 4th value, which is 57.
- Range: Subtract the smallest value from the largest value. The range is 61 - 57 = 4.

2. For the second data set (second 7 presidents):
- Mean: Add up all the numbers (61 + 52 + 69 + 64 + 46 + 54 + 47) and divide by the total count (7). The mean is 56.
- Median: Arrange the numbers in ascending order (46, 47, 52, 54, 61, 64, 69) and find the middle value. Since there are 7 numbers, the median is the 4th value, which is 54.
- Range: Subtract the smallest value from the largest value. The range is 69 - 46 = 23.

b. Finding the Standard Deviation:

To find the standard deviation using the rule of thumb, we need to calculate the average deviation from the mean.

1. For the first data set:
- Calculate the deviations: Subtract the mean (58) from each number, and square the result. The deviations are: (-1)^2, (3)^2, (-1)^2, (-1)^2, (0)^2, (-1)^2, (3)^2.
- Sum the squared deviations: Add up all the squared deviations.
- Divide by the count: Since we have 7 numbers, divide the sum of squared deviations by 7.
- Take the square root: Find the square root of the result to get the standard deviation.

2. For the second data set:
- Calculate the deviations: Subtract the mean (56) from each number, and square the result.
- Sum the squared deviations: Add up all the squared deviations.
- Divide by the count: Since we have 7 numbers, divide the sum of squared deviations by 7.
- Take the square root: Find the square root of the result to get the standard deviation.

c. Comparing the Two Sets:

After finding the mean, median, range, and standard deviation for both sets, compare the results and describe the differences or similarities between the two sets. Look for patterns, differences in variability, or any other observations that can be made.

We do not do your work for you. Once you have answered your questions, we will be happy to give you feedback on your work. Although it might require more time and effort, you will learn more if you do your own work. Isn't that why you go to school?

Range = highest score - lowest

Median = 50th percentile. Arrange scores in order of value. Middle-most value = median. If two middle-most values, find mean of the two middle-most values.

Don't know what rule of thumb you mean.

For standard deviation, find the mean first = sum of scores/number of scores

Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.

Standard deviation = square root of variance

I'll let you do the calculations.