Answer :
Sure, let's find the standard deviation of the given data step-by-step.
### Step 1: Understand the Data
We have the following mileage ranges and their frequencies:
| Miles Ran | Frequency |
|--------------------|-----------|
| 5001 - 20000 | 1 |
| 20001 - 45000 | 3 |
| 45001 - 60000 | 12 |
| 60001 - 75000 | 13 |
| 75001 - 90000 | 19 |
| 90001 - 105000 | 26 |
### Step 2: Calculate the Midpoints
For each mileage range, calculate the midpoint (average of the lower and upper bounds):
- Midpoint of 5001 - 20000 = (5001 + 20000) / 2 = 12500.5
- Midpoint of 20001 - 45000 = (20001 + 45000) / 2 = 32500.5
- Midpoint of 45001 - 60000 = (45001 + 60000) / 2 = 52500.5
- Midpoint of 60001 - 75000 = (60001 + 75000) / 2 = 67500.5
- Midpoint of 75001 - 90000 = (75001 + 90000) / 2 = 82500.5
- Midpoint of 90001 - 105000 = (90001 + 105000) / 2 = 97500.5
### Step 3: Calculate the Mean
To find the mean (average) of the data, use the following formula:
[tex]\[ \text{Mean} = \frac{\sum (\text{midpoint} \times \text{frequency})}{\sum \text{frequency}} \][/tex]
First, calculate the total frequency:
[tex]\[ \text{Total frequency} = 1 + 3 + 12 + 13 + 19 + 26 = 74 \][/tex]
Next, calculate the sum of the products of midpoints and frequencies:
[tex]\[ \sum (\text{midpoint} \times \text{frequency}) = 12500.5 \times 1 + 32500.5 \times 3 + 52500.5 \times 12 + 67500.5 \times 13 + 82500.5 \times 19 + 97500.5 \times 26 \][/tex]
[tex]\[ = 12500.5 + 97501.5 + 630006 + 877506.5 + 1568509.5 + 2532013 = 5718036 \][/tex]
Now, calculate the mean:
[tex]\[ \text{Mean} = \frac{5718036}{74} = 77297.8 \][/tex]
### Step 4: Calculate the Variance
Variance is calculated using the formula:
[tex]\[ \text{Variance} = \frac{\sum (\text{frequency} \times (\text{midpoint} - \text{mean})^2)}{\sum \text{frequency}} \][/tex]
Calculate each term of the sum:
[tex]\[ (12500.5 - 77297.8)^2 \times 1 = 4188619471.2 \][/tex]
[tex]\[ (32500.5 - 77297.8)^2 \times 3 = 6702683131.3 \][/tex]
[tex]\[ (52500.5 - 77297.8)^2 \times 12 = 6157698756.8 \][/tex]
[tex]\[ (67500.5 - 77297.8)^2 \times 13 = 1300982134.9 \][/tex]
[tex]\[ (82500.5 - 77297.8)^2 \times 19 = 439510888.5 \][/tex]
[tex]\[ (97500.5 - 77297.8)^2 \times 26 = 1126357096.3 \][/tex]
Sum these values:
[tex]\[ 4188619471.2 + 6702683131.3 + 6157698756.8 + 1300982134.9 + 439510888.5 + 1126357096.3 = 29971958479.2 \][/tex]
Calculate the variance:
[tex]\[ \text{Variance} = \frac{29971958479.2}{74} = 405026479.2 \][/tex]
### Step 5: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
[tex]\[ \text{Standard Deviation} = \sqrt{405026479.2} \approx 20125 \][/tex]
### Final Answer
The standard deviation of the data is approximately [tex]\( \boxed{20125} \)[/tex].
### Step 1: Understand the Data
We have the following mileage ranges and their frequencies:
| Miles Ran | Frequency |
|--------------------|-----------|
| 5001 - 20000 | 1 |
| 20001 - 45000 | 3 |
| 45001 - 60000 | 12 |
| 60001 - 75000 | 13 |
| 75001 - 90000 | 19 |
| 90001 - 105000 | 26 |
### Step 2: Calculate the Midpoints
For each mileage range, calculate the midpoint (average of the lower and upper bounds):
- Midpoint of 5001 - 20000 = (5001 + 20000) / 2 = 12500.5
- Midpoint of 20001 - 45000 = (20001 + 45000) / 2 = 32500.5
- Midpoint of 45001 - 60000 = (45001 + 60000) / 2 = 52500.5
- Midpoint of 60001 - 75000 = (60001 + 75000) / 2 = 67500.5
- Midpoint of 75001 - 90000 = (75001 + 90000) / 2 = 82500.5
- Midpoint of 90001 - 105000 = (90001 + 105000) / 2 = 97500.5
### Step 3: Calculate the Mean
To find the mean (average) of the data, use the following formula:
[tex]\[ \text{Mean} = \frac{\sum (\text{midpoint} \times \text{frequency})}{\sum \text{frequency}} \][/tex]
First, calculate the total frequency:
[tex]\[ \text{Total frequency} = 1 + 3 + 12 + 13 + 19 + 26 = 74 \][/tex]
Next, calculate the sum of the products of midpoints and frequencies:
[tex]\[ \sum (\text{midpoint} \times \text{frequency}) = 12500.5 \times 1 + 32500.5 \times 3 + 52500.5 \times 12 + 67500.5 \times 13 + 82500.5 \times 19 + 97500.5 \times 26 \][/tex]
[tex]\[ = 12500.5 + 97501.5 + 630006 + 877506.5 + 1568509.5 + 2532013 = 5718036 \][/tex]
Now, calculate the mean:
[tex]\[ \text{Mean} = \frac{5718036}{74} = 77297.8 \][/tex]
### Step 4: Calculate the Variance
Variance is calculated using the formula:
[tex]\[ \text{Variance} = \frac{\sum (\text{frequency} \times (\text{midpoint} - \text{mean})^2)}{\sum \text{frequency}} \][/tex]
Calculate each term of the sum:
[tex]\[ (12500.5 - 77297.8)^2 \times 1 = 4188619471.2 \][/tex]
[tex]\[ (32500.5 - 77297.8)^2 \times 3 = 6702683131.3 \][/tex]
[tex]\[ (52500.5 - 77297.8)^2 \times 12 = 6157698756.8 \][/tex]
[tex]\[ (67500.5 - 77297.8)^2 \times 13 = 1300982134.9 \][/tex]
[tex]\[ (82500.5 - 77297.8)^2 \times 19 = 439510888.5 \][/tex]
[tex]\[ (97500.5 - 77297.8)^2 \times 26 = 1126357096.3 \][/tex]
Sum these values:
[tex]\[ 4188619471.2 + 6702683131.3 + 6157698756.8 + 1300982134.9 + 439510888.5 + 1126357096.3 = 29971958479.2 \][/tex]
Calculate the variance:
[tex]\[ \text{Variance} = \frac{29971958479.2}{74} = 405026479.2 \][/tex]
### Step 5: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
[tex]\[ \text{Standard Deviation} = \sqrt{405026479.2} \approx 20125 \][/tex]
### Final Answer
The standard deviation of the data is approximately [tex]\( \boxed{20125} \)[/tex].
