How do I estimate the mean and estimate the median of this table?
|Amount|
_____________
0-8
8-16
16-24
______________
|Frequency|
______________
18
15
16
_______________
|Amount|
_______________
24-32
32-40
40-48
_______________
|Frequency|
_______________
11
9
6
_______________
To estimate the mean and median of the given table, you would need to follow these steps:
1. Calculate the midpoint of each class interval: To find the midpoint, add the lower and the upper limit of each class interval and divide the sum by 2.
For the first table:
- Midpoint for the first interval (0-8): (0 + 8) / 2 = 4
- Midpoint for the second interval (8-16): (8 + 16) / 2 = 12
- Midpoint for the third interval (16-24): (16 + 24) / 2 = 20
For the second table:
- Midpoint for the fourth interval (24-32): (24 + 32) / 2 = 28
- Midpoint for the fifth interval (32-40): (32 + 40) / 2 = 36
- Midpoint for the sixth interval (40-48): (40 + 48) / 2 = 44
2. Calculate the product of the midpoint and the frequency for each interval: Multiply the midpoint of each interval with its corresponding frequency.
For the first table:
- Product for the first interval: 4 * 18 = 72
- Product for the second interval: 12 * 15 = 180
- Product for the third interval: 20 * 16 = 320
For the second table:
- Product for the fourth interval: 28 * 11 = 308
- Product for the fifth interval: 36 * 9 = 324
- Product for the sixth interval: 44 * 6 = 264
3. Calculate the total of the products: Add up all the products calculated in the previous step.
For the first table: 72 + 180 + 320 = 572
For the second table: 308 + 324 + 264 = 896
4. Calculate the total frequency: Add up all the frequencies provided in the table.
For the first table: 18 + 15 + 16 = 49
For the second table: 11 + 9 + 6 = 26
5. Estimate the mean: Divide the total product by the total frequency.
For the first table: 572 / 49 = 11.67 (estimated mean)
For the second table: 896 / 26 = 34.46 (estimated mean)
6. Estimate the median: The median is the value that divides the dataset into two equal halves. To estimate the median, we need to find the cumulative frequency.
For the first table:
- The cumulative frequency for the first interval (0-8): 18
- The cumulative frequency for the second interval (0-16): 18 + 15 = 33
Since the cumulative frequency is less than half of the total frequency, the median lies within the second interval (8-16). To find the estimated median, we can use interpolation.
Interpolated median = Lower limit of the interval + [(N/2 - cumulative frequency of the previous interval) / frequency of the current interval] * class interval width
Interpolated median = 8 + [(49/2 - 18) / 15] * 8 = 8 + [15.33/15] * 8 = 8 + 10.22 = 18.22 (estimated median)
For the second table:
- The cumulative frequency for the fourth interval (0-32): 11 + 9 = 20
- The cumulative frequency for the fifth interval (0-40): 11 + 9 + 6 = 26
Since the cumulative frequency equals the total frequency, the median lies within the fifth interval (32-40). Using the interpolation formula:
Interpolated median = Lower limit of the interval + [(N/2 - cumulative frequency of the previous interval) / frequency of the current interval] * class interval width
Interpolated median = 32 + [(26/2 - 20) / 9] * 8 = 32 + [0.67/9] * 8 = 32 + 0.59 = 32.59 (estimated median)
Thus, the estimated mean and median for the given tables are as follows:
First Table:
- Estimated mean: 11.67
- Estimated median: 18.22
Second Table:
- Estimated mean: 34.46
- Estimated median: 32.59