1. For the following exercise, complete the following by using the data sets below that provides the ages of the first seven presidents:
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 (please see Chapter 4, Section 4.3 of the text).
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
2. A data set consists of a set of numerical values. Please indicate by stating yes or no for each of the following statements whether or not it could be correct.
a. There is no mode.
b. There are two modes.
c. There are three modes.
3. Indicate whether the given statement could apply to a data set consisting of 1,000 values that are all different.
a. The 29th percentile is greater than the 30th percentile.
b. The median is greater than the first quartile.
c. The third quartile is greater than the first quartile.
d. The mean is equal to the median.
e. The range is zero.
First, we do not do your work for you. Second, we do not have your text. You will learn more if you calculate the mean, median, mode and standard deviations yourself.
However, I will tell you that the answer to 3 is b and c.
the answer to 2 is a
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
1. To find the mean, median, and range for each data set, follow these steps:
a. Mean:
- Add up all the values in the data set.
- Divide the sum by the total number of values in the data set (in this case, 7 for each set).
- The result is the mean.
For the first data set:
Mean = (57 + 61 + 57 + 57 + 58 + 57 + 61) / 7 = 415 / 7 = 59.29
For the second data set:
Mean = (61 + 52 + 69 + 64 + 46 + 54 + 47) / 7 = 393 / 7 = 56.14
b. Median:
- Arrange the values in the data set in ascending order.
- If the total number of values is odd, the median is the middle value.
- If the total number of values is even, the median is the average of the two middle values.
For the first data set:
Arranged in ascending order: 57, 57, 57, 58, 61, 61
Median = 57
For the second data set:
Arranged in ascending order: 46, 47, 52, 54, 61, 64, 69
Median = (54 + 61) / 2 = 115 / 2 = 57.5
c. Range:
- Calculate the difference between the maximum value and the minimum value in the data set.
For the first data set:
Range = 61 - 57 = 4
For the second data set:
Range = 69 - 46 = 23
b. To find the standard deviation using the rule of thumb, follow these steps:
- Find the mean of the data set (already calculated).
- Subtract the mean from each value in the data set to get the deviations.
- Square each deviation.
- Find the average (mean) of the squared deviations.
- Take the square root of the result.
For the first data set:
Deviations: -2.29, 1.71, -2.29, -2.29, -1.29, -2.29, 1.71
Squared deviations: 5.2441, 2.9241, 5.2441, 5.2441, 1.6641, 5.2441, 2.9241
Average of squared deviations: (5.2441 + 2.9241 + 5.2441 + 5.2441 + 1.6641 + 5.2441 + 2.9241) / 7 = 3.947
Standard deviation = √3.947 ≈ 1.986 (rounded to three decimal places)
For the second data set:
Deviations: 4.86, -4.14, 12.86, 7.86, -10.14, -2.14, -9.14
Squared deviations: 23.6196, 17.1396, 165.6196, 61.5396, 102.8196, 4.5796, 83.6196
Average of squared deviations: (23.6196 + 17.1396 + 165.6196 + 61.5396 + 102.8196 + 4.5796 + 83.6196) / 7 = 64.201
Standard deviation = √64.201 ≈ 8.010 (rounded to three decimal places)
c. To compare the two sets, observe the following:
- Mean: The mean of the first set is 59.29, while the mean of the second set is 56.14. This indicates that, on average, the presidents in the first set had a slightly higher age at inauguration.
- Median: The median of the first set is 57, while the median of the second set is 57.5. This suggests that the middle-age presidents in both sets had similar ages.
- Range: The range of the first set is 4, while the range of the second set is 23. This shows that the spread or variability of ages in the second set is larger than in the first set.
- Standard Deviation: The standard deviation of the first set is 1.986, while the standard deviation of the second set is 8.010. This further indicates that the second set has a larger spread or variability.
In summary, the first set of presidents had a slightly higher average age at inauguration, a smaller range, and a smaller standard deviation compared to the second set of presidents.
2. For the statements about modes in a data set:
a. Yes, there can be no mode if all the values in the data set occur with the same frequency.
b. Yes, there can be two modes if two (or more) values have the highest frequency in the data set.
c. Yes, there can be three modes if three (or more) values have the same highest frequency in the data set.
3. For the given statements about a data set of 1,000 different values:
a. No, the 29th percentile cannot be greater than the 30th percentile. Percentiles are arranged in increasing order, so the higher percentile values must be greater than the lower percentile values.
b. Yes, the median can be greater than the first quartile. The first quartile represents the 25th percentile, while the median represents the 50th percentile. The data set can have values that are higher in the second half (50th percentile) compared to the first half (25th percentile).
c. Yes, the third quartile can be greater than the first quartile. The third quartile represents the 75th percentile, while the first quartile represents the 25th percentile. The data set can have values that are higher in the upper half (75th percentile) compared to the lower half (25th percentile).
d. No, in a data set with 1,000 different values, the mean and median will not necessarily be equal. The mean is influenced by the sum of all the values, while the median only looks at the middle value(s) in the data set.
e. Yes, the range can be zero if all the values in the data set are exactly the same. The range is calculated as the difference between the maximum and minimum values, which would be zero in this case.