Given the following frequency distribution, find the mean, variance, and standard deviation. Please show all of your work.
Errors Frequency
51-53 9
54-56 19
57-59 16
60-62 24
63-65 25
This is what I got:
Mean = 59.19
Variance = 1.43
Standard Deviation = 2.05
To find the mean, variance, and standard deviation, you need to follow these steps:
Step 1: Calculate the midpoint of each class interval.
The midpoint is found by adding the lower and upper limits of each interval and dividing by 2.
For example:
- The midpoint of the interval 51-53 is (51 + 53) / 2 = 52
- The midpoint of the interval 54-56 is (54 + 56) / 2 = 55
- And so on...
Step 2: Multiply each midpoint by its corresponding frequency.
For example:
- Multiply the midpoint of 51-53 (52) by its frequency (9): 52 * 9 = 468
- Multiply the midpoint of 54-56 (55) by its frequency (19): 55 * 19 = 1,045
- And so on...
Step 3: Calculate the sum of all the products obtained in step 2.
For example:
- Sum the products obtained in the previous step: 468 + 1,045 + ... = ...
Step 4: Calculate the sum of the frequencies.
For example:
- Sum all frequencies: 9 + 19 + ... = ...
Step 5: Calculate the mean.
The mean is obtained by dividing the sum of the products (step 3) by the sum of the frequencies (step 4).
For example:
- Divide the sum of products by the sum of frequencies: (sum of products) / (sum of frequencies) = mean
Step 6: Calculate the variance.
The variance is found by calculating the sum of the squares of the differences between each midpoint and the mean, multiplied by their respective frequencies. Then, divide this sum by the sum of the frequencies.
For example:
- Calculate the squared difference between each midpoint and the mean, multiply by the frequency, and sum the products: ((52 - mean)^2 * 9) + ((55 - mean)^2 * 19) + ... = ...
- Divide the result by the sum of frequencies: variance = (sum of products) / (sum of frequencies)
Step 7: Calculate the standard deviation.
The standard deviation is the square root of the variance.
For example:
- Calculate the square root of the variance obtained in step 6: standard deviation = sqrt(variance)
By following these steps, you should be able to calculate the mean, variance, and standard deviation using the given frequency distribution.