Find the mean deviation about the mean of the following data of ages of married men in a certain town.
Ages 15-24:25-34 35-44 45-54 55-64 65-74
No. of Men 33 264 303 214 128 58
Mean = (33*15 + 264*25 + 303*35 + 214*45 + 128*55 + 58*65)/(33 + 264 + 303 + 214 + 128 + 58)
Mean = 39.7
Mean Deviation about the Mean = (33*|15-39.7| + 264*|25-39.7| + 303*|35-39.7| + 214*|45-39.7| + 128*|55-39.7| + 58*|65-39.7|)/(33 + 264 + 303 + 214 + 128 + 58)
Mean Deviation about the Mean = 17.7
Mean = (33*15 + 264*25 + 303*35 + 214*45 + 128*55 + 58*65)/(33 + 264 + 303 + 214 + 128 + 58)
Mean = 39.7
Mean Deviation about the Mean = (33*|15-39.7| + 264*|25-39.7| + 303*|35-39.7| + 214*|45-39.7| + 128*|55-39.7| + 58*|65-39.7|)/(33 + 264 + 303 + 214 + 128 + 58)
Mean Deviation about the Mean = 17.7
To find the mean deviation about the mean, we need to follow these steps:
1. Calculate the mean:
- Multiply each age range by its respective number of men:
(20 + 30) * 33 = 1056
(30 + 40) * 264 = 13056
(40 + 50) * 303 = 30330
(50 + 60) * 214 = 21400
(60 + 70) * 128 = 14080
(70 + 80) * 58 = 7540
- Add all the products:
1056 + 13056 + 30330 + 21400 + 14080 + 7540 = 88,462
- Divide the sum by the total number of men:
88,462 ÷ (33 + 264 + 303 + 214 + 128 + 58) = 88,462 ÷ 1000 = 88.462
The mean of the ages is approximately 88.462.
2. Calculate the deviation for each age range:
- Subtract the mean from each age range multiplied by its respective number of men:
(20 + 30) * 33 - 88.462 = 1,056 - 88.462 = 967.538
(30 + 40) * 264 - 88.462 = 13,056 - 88.462 = 12,967.538
(40 + 50) * 303 - 88.462 = 30,330 - 88.462 = 30,241.538
(50 + 60) * 214 - 88.462 = 21,400 - 88.462 = 21,311.538
(60 + 70) * 128 - 88.462 = 14,080 - 88.462 = 13,991.538
(70 + 80) * 58 - 88.462 = 7,540 - 88.462 = 7,451.538
3. Calculate the absolute value of each deviation:
- Disregard any negative signs to get the absolute values:
967.538
12,967.538
30,241.538
21,311.538
13,991.538
7,451.538
4. Sum all the absolute deviations:
967.538 + 12,967.538 + 30,241.538 + 21,311.538 + 13,991.538 + 7,451.538 = 86,931.728
5. Calculate the mean deviation about the mean:
- Divide the sum of absolute deviations by the total number of men:
86,931.728 ÷ 1000 = 86.931728
Therefore, the mean deviation about the mean of the data is approximately 86.931728.
To find the mean deviation about the mean, follow these steps:
1. Calculate the mean of the data.
To calculate the mean, multiply each age group by the corresponding number of men, and then divide the result by the total number of men (sum of all the number of men):
Mean = ((15-24) * 33 + (25-34) * 264 + (35-44) * 303 + (45-54) * 214 + (55-64) * 128 + (65-74) * 58) / (33 + 264 + 303 + 214 + 128 + 58)
2. Subtract the mean from each age, and then find the absolute value of the difference.
For each age group, subtract the mean calculated in step 1 from the center of the age group (the average of the upper and lower limits). Then, take the absolute value of the difference.
For example, for the age group 15-24:
- Subtract the mean from the center of the age group: (24 + 15) / 2 - calculated mean
- Take the absolute value of the difference: |(24 + 15) / 2 - calculated mean|
Repeat this step for each age group.
3. Multiply each absolute difference by the corresponding number of men.
For each age group, multiply the absolute difference calculated in step 2 by the number of men in that age group.
For example, for the age group 15-24:
- Multiply the absolute difference by the number of men: |(24 + 15) / 2 - calculated mean| * 33
Repeat this step for each age group.
4. Find the sum of all the products calculated in step 3.
Add up all the products calculated in step 3 to find the sum.
5. Divide the sum from step 4 by the total number of men.
Divide the sum from step 4 by the total number of men to get the mean deviation about the mean.
Once you perform these calculations, you will find the mean deviation about the mean of the given data of ages of married men in the town.