Answer :
Sure! Let's go through the steps to solve this question. We are tasked with finding the range and the standard deviation of a sample data set.
### Given Data
The sample data set is:
- 59.9, 63.9, 78.2, 97.3, 76.2, 83.3, 97.3, 60.2, 97.3, 73.8
### Step 1: Calculate the Range
The range of a data set is the difference between the maximum value and the minimum value.
1. Find the maximum value: 97.3
2. Find the minimum value: 59.9
3. Calculate the range:
[tex]\[
\text{Range} = \text{Maximum} - \text{Minimum} = 97.3 - 59.9 = 37.4
\][/tex]
So, the range of this data set is 37.4.
### Step 2: Calculate the Sample Standard Deviation
The sample standard deviation measures how spread out the numbers in the data set are from the sample mean. It is calculated as follows:
1. Find the mean (average) of the data set.
[tex]\[
\text{Mean} = \frac{59.9 + 63.9 + 78.2 + 97.3 + 76.2 + 83.3 + 97.3 + 60.2 + 97.3 + 73.8}{10}
\][/tex]
2. Find the squared deviation of each data point from the mean, and then average those squared deviations.
3. Take the square root of that average. This is the sample standard deviation.
For the given data set, the sample standard deviation is calculated to be approximately 14.92 when rounded to two decimal places more than the original data.
These calculations show that the range is 37.4 and the sample standard deviation is approximately 14.92.
### Given Data
The sample data set is:
- 59.9, 63.9, 78.2, 97.3, 76.2, 83.3, 97.3, 60.2, 97.3, 73.8
### Step 1: Calculate the Range
The range of a data set is the difference between the maximum value and the minimum value.
1. Find the maximum value: 97.3
2. Find the minimum value: 59.9
3. Calculate the range:
[tex]\[
\text{Range} = \text{Maximum} - \text{Minimum} = 97.3 - 59.9 = 37.4
\][/tex]
So, the range of this data set is 37.4.
### Step 2: Calculate the Sample Standard Deviation
The sample standard deviation measures how spread out the numbers in the data set are from the sample mean. It is calculated as follows:
1. Find the mean (average) of the data set.
[tex]\[
\text{Mean} = \frac{59.9 + 63.9 + 78.2 + 97.3 + 76.2 + 83.3 + 97.3 + 60.2 + 97.3 + 73.8}{10}
\][/tex]
2. Find the squared deviation of each data point from the mean, and then average those squared deviations.
3. Take the square root of that average. This is the sample standard deviation.
For the given data set, the sample standard deviation is calculated to be approximately 14.92 when rounded to two decimal places more than the original data.
These calculations show that the range is 37.4 and the sample standard deviation is approximately 14.92.
