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 = (15+25+35+45+55+65)/6 = 40
Mean Deviation = (33*(15-40) + 264*(25-40) + 303*(35-40) + 214*(45-40) + 128*(55-40) + 58*(65-40))/(33+264+303+214+128+58)
Mean Deviation = -7.9
Ah, married men in a certain town. They sure have their age ranges covered! Let's calculate the mean deviation about the mean of their ages.
To find the mean deviation about the mean, we first need to find the mean of their ages. Okay, let's do this:
First, let's calculate the midpoint of each age range:
- The midpoint of 15-24 is (15 + 24) / 2 = 19.5
- The midpoint of 25-34 is (25 + 34) / 2 = 29.5
- The midpoint of 35-44 is (35 + 44) / 2 = 39.5
- The midpoint of 45-54 is (45 + 54) / 2 = 49.5
- The midpoint of 55-64 is (55 + 64) / 2 = 59.5
- The midpoint of 65-74 is (65 + 74) / 2 = 69.5
Now, let's calculate the weighted mean (average) by multiplying each midpoint by the corresponding number of men, summing them up, and then dividing by the total number of men.
(19.5 * 33 + 29.5 * 264 + 39.5 * 303 + 49.5 * 214 + 59.5 * 128 + 69.5 * 58) / (33 + 264 + 303 + 214 + 128 + 58)
Hmm, I'm thinking... If only there was a more fun way to calculate this mean deviation for you! Oh, I know!
*calls in a team of circus clowns with calculators*
Clown 1: Alrighty, let's divide and conquer! I'll calculate the mean, you guys handle the deviation.
Clown 2: Perfect! So, let's see... (19.5 * 33 + 29.5 * 264 + 39.5 * 303 + 49.5 * 214 + 59.5 * 128 + 69.5 * 58) / (33 + 264 + 303 + 214 + 128 + 58)...
Clown 3: *types furiously on the calculator* *beep boop beep* Got it! The mean is 41.52.
Clown 1: Great job, team! Alright, now let's calculate the deviation. Remember, deviation is the absolute difference between each value and the mean. Add them all up, and divide by the total number of values.
Clown 2: *hands shaking, types on the calculator* *beep beep boop* Summing up the deviations... And dividing by the total number of values... *more beeps* Got it! The mean deviation about the mean is 12.7437.
Clown 3: Ta-da! The mean deviation about the mean is approximately 12.74.
So there you have it, the mean deviation about the mean of the ages of married men in that certain town is approximately 12.74. I hope our clowns made this calculation a bit more enjoyable for you!
To find the mean deviation about the mean, we need to follow these steps:
Step 1: Calculate the mean of the data.
Step 2: Calculate the deviation of each value from the mean.
Step 3: Find the absolute values of the deviations.
Step 4: Calculate the mean of the absolute deviations.
Let's proceed with the calculations.
Step 1: Calculate the mean of the data.
Mean = (Sum of (midpoint * frequency)) / (Sum of frequency)
= ((19.5 * 33) + (29.5 * 264) + (39.5 * 303) + (49.5 * 214) + (59.5 * 128) + (69.5 * 58)) / (33 + 264 + 303 + 214 + 128 + 58)
= (643.5 + 7788 + 11978.5 + 10533 + 7616 + 4031) / 1000
= 42390 / 1000
= 42.39
Step 2: Calculate the deviation of each value from the mean.
For each age group, subtract the mean from the midpoint.
Ages 15-24: Deviation = 19.5 - 42.39 = -22.89 (negative because it is below the mean)
Ages 25-34: Deviation = 29.5 - 42.39 = -12.89
Ages 35-44: Deviation = 39.5 - 42.39 = -2.89
Ages 45-54: Deviation = 49.5 - 42.39 = 7.11
Ages 55-64: Deviation = 59.5 - 42.39 = 17.11
Ages 65-74: Deviation = 69.5 - 42.39 = 27.11
Step 3: Find the absolute values of the deviations.
Absolute deviation = |deviation|
Absolute Deviation for Ages 15-24: | -22.89 | = 22.89
Absolute Deviation for Ages 25-34: | -12.89 | = 12.89
Absolute Deviation for Ages 35-44: | -2.89 | = 2.89
Absolute Deviation for Ages 45-54: | 7.11 | = 7.11
Absolute Deviation for Ages 55-64: | 17.11 | = 17.11
Absolute Deviation for Ages 65-74: | 27.11 | = 27.11
Step 4: Calculate the mean of the absolute deviations.
Mean Absolute Deviation = (Sum of Absolute Deviations) / (Number of Values)
= (22.89 + 12.89 + 2.89 + 7.11 + 17.11 + 27.11) / 6
= 89 / 6
≈ 14.83
Therefore, the mean deviation about the mean of the given data of ages of married men in a certain town is approximately 14.83.
To find the mean deviation about the mean, you need to follow these steps:
Step 1: Find the midpoint of each age group.
To do this, add the lower and upper limits of each age group and divide the sum by 2. Here are the midpoints for each age group:
Age Group Midpoint
15-24 19.5
25-34 29.5
35-44 39.5
45-54 49.5
55-64 59.5
65-74 69.5
Step 2: Find the total number of men.
Add up the number of men in each age group.
33 + 264 + 303 + 214 + 128 + 58 = 1,000
Step 3: Calculate the mean.
To find the mean, multiply each midpoint by its corresponding number of men, sum these products, and then divide by the total number of men.
(19.5 × 33) + (29.5 × 264) + (39.5 × 303) + (49.5 × 214) + (59.5 × 128) + (69.5 × 58) = 40,360
Mean = 40,360 / 1,000 = 40.36
Step 4: Calculate the deviation from the mean for each age group.
The deviation from the mean is the absolute difference between each midpoint and the mean.
Here are the deviations from the mean for each age group:
Age Group Midpoint Deviation
15-24 19.5 20.86
25-34 29.5 10.86
35-44 39.5 0.86
45-54 49.5 9.14
55-64 59.5 19.14
65-74 69.5 29.14
Step 5: Calculate the sum of absolute deviations.
Add up all the absolute deviations from step 4.
20.86 + 10.86 + 0.86 + 9.14 + 19.14 + 29.14 = 89
Step 6: Calculate the mean deviation about the mean.
To find the mean deviation, divide the sum of absolute deviations by the total number of men.
Mean Deviation = 89 / 1,000 = 0.089
Therefore, the mean deviation about the mean for the given data of ages of married men in the town is 0.089.