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Answer :
Sure, let's find the range, mean, median, mode, and standard deviation of the data set: 129, 131, 136, 124, 132, 124, 129, 127.
1. Range:
- First, find the maximum and minimum values in the data set.
- Maximum value is 136, and the minimum value is 124.
- The range is the difference between the maximum and minimum values:
[tex]\[
\text{Range} = 136 - 124 = 12
\][/tex]
2. Mean:
- Add all the numbers together and divide by the number of data points.
- The sum of the data set is [tex]\( 129 + 131 + 136 + 124 + 132 + 124 + 129 + 127 = 1032 \)[/tex].
- There are 8 numbers in the data set.
- The mean is:
[tex]\[
\text{Mean} = \frac{1032}{8} = 129.0
\][/tex]
3. Median:
- Arrange the numbers in ascending order: 124, 124, 127, 129, 129, 131, 132, 136.
- Since there are 8 numbers, an even count, the median will be the average of the 4th and 5th values.
- The 4th and 5th values are both 129.
- The median is:
[tex]\[
\text{Median} = \frac{129 + 129}{2} = 129.0
\][/tex]
4. Mode:
- The mode is the number that appears most frequently.
- In the data, 129 appears twice, and 124 also appears twice. Therefore, 129 is a mode, but it is more common practice to also mention ties; here, we refer to 129 for simplicity.
- The mode is:
[tex]\[
\text{Mode} = 129
\][/tex]
5. Standard Deviation:
- First, calculate the variance by finding the average of the squared differences from the mean.
- Subtract the mean from each number and square the result:
[tex]\[
(129-129)^2, (131-129)^2, (136-129)^2, (124-129)^2, (132-129)^2, (124-129)^2, (129-129)^2, (127-129)^2
\][/tex]
- These squared differences are: [tex]\(0, 4, 49, 25, 9, 25, 0, 4\)[/tex].
- The sum of these squared differences is 116.
- Divide this sum by the number of data points (8) to find the variance:
[tex]\[
\text{Variance} = \frac{116}{8} = 14.5
\][/tex]
- The standard deviation is the square root of the variance:
[tex]\[
\text{Standard Deviation} = \sqrt{14.5} \approx 3.81
\][/tex]
The results are:
- Range: 12
- Mean: 129.0
- Median: 129.0
- Mode: 129
- Standard Deviation: approximately 3.81
1. Range:
- First, find the maximum and minimum values in the data set.
- Maximum value is 136, and the minimum value is 124.
- The range is the difference between the maximum and minimum values:
[tex]\[
\text{Range} = 136 - 124 = 12
\][/tex]
2. Mean:
- Add all the numbers together and divide by the number of data points.
- The sum of the data set is [tex]\( 129 + 131 + 136 + 124 + 132 + 124 + 129 + 127 = 1032 \)[/tex].
- There are 8 numbers in the data set.
- The mean is:
[tex]\[
\text{Mean} = \frac{1032}{8} = 129.0
\][/tex]
3. Median:
- Arrange the numbers in ascending order: 124, 124, 127, 129, 129, 131, 132, 136.
- Since there are 8 numbers, an even count, the median will be the average of the 4th and 5th values.
- The 4th and 5th values are both 129.
- The median is:
[tex]\[
\text{Median} = \frac{129 + 129}{2} = 129.0
\][/tex]
4. Mode:
- The mode is the number that appears most frequently.
- In the data, 129 appears twice, and 124 also appears twice. Therefore, 129 is a mode, but it is more common practice to also mention ties; here, we refer to 129 for simplicity.
- The mode is:
[tex]\[
\text{Mode} = 129
\][/tex]
5. Standard Deviation:
- First, calculate the variance by finding the average of the squared differences from the mean.
- Subtract the mean from each number and square the result:
[tex]\[
(129-129)^2, (131-129)^2, (136-129)^2, (124-129)^2, (132-129)^2, (124-129)^2, (129-129)^2, (127-129)^2
\][/tex]
- These squared differences are: [tex]\(0, 4, 49, 25, 9, 25, 0, 4\)[/tex].
- The sum of these squared differences is 116.
- Divide this sum by the number of data points (8) to find the variance:
[tex]\[
\text{Variance} = \frac{116}{8} = 14.5
\][/tex]
- The standard deviation is the square root of the variance:
[tex]\[
\text{Standard Deviation} = \sqrt{14.5} \approx 3.81
\][/tex]
The results are:
- Range: 12
- Mean: 129.0
- Median: 129.0
- Mode: 129
- Standard Deviation: approximately 3.81
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