let the first entire population be -10 0 10 20 30 and second 8 9 10 11 12 calculate the mean, median, mode range variance and Q also compare they mean
To calculate the mean, median, mode, range, variance, and Q (lower quartile), follow these steps:
Step 1: Calculate the mean:
Mean = (sum of all values) / (number of values)
For the first population:
Mean = (-10 + 0 + 10 + 20 + 30) / 5 = 50 / 5 = 10
For the second population:
Mean = (8 + 9 + 10 + 11 + 12) / 5 = 50 / 5 = 10
So both populations have a mean of 10.
Step 2: Calculate the median:
Arranging the values in ascending order:
First population: -10, 0, 10, 20, 30.
Median = (10 + 10) / 2 = 10
Second population: 8, 9, 10, 11, 12.
Median = 10
Both populations have a median of 10.
Step 3: Calculate the mode:
Mode is the value(s) that appears most frequently.
First population: No mode as all values appear only once.
Second population: Mode = 10 (appears the most frequently)
Only the second population has a mode, which is 10.
Step 4: Calculate the range:
Range = (highest value) - (lowest value)
For the first population: 30 - (-10) = 40
For the second population: 12 - 8 = 4
The range of the first population is 40, and the range of the second population is 4.
Step 5: Calculate the variance:
Variance measures the dispersion of data points.
For the first population:
Variance = (sum of squared differences from the mean) / (number of values)
Variance = [(-10-10)^2 + (0-10)^2 + (10-10)^2 + (20-10)^2 + (30-10)^2] / 5
= [(-20)^2 + (-10)^2 + (0)^2 + (10)^2 + (20)^2] / 5
= [400 + 100 + 0 + 100 + 400] / 5
= 1000 / 5
= 200
For the second population:
Variance = [(8-10)^2 + (9-10)^2 + (10-10)^2 + (11-10)^2 + (12-10)^2] / 5
= [(-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2] / 5
= [4 + 1 + 0 + 1 + 4] / 5
= 10 / 5
= 2
The variance of the first population is 200, while the variance of the second population is 2.
Step 6: Calculate the lower quartile (Q):
Q is the median of the lower half of the data.
For the first population: Q = -10
For the second population: Q = 9
The first population has a lower quartile of -10, while the second population has a lower quartile of 9.
To compare the means:
Both populations have the same mean, which is 10. Therefore, the means are equal.
In summary:
For the first population:
Mean: 10
Median: 10
Mode: No mode
Range: 40
Variance: 200
Lower Quartile (Q): -10
For the second population:
Mean: 10
Median: 10
Mode: 10
Range: 4
Variance: 2
Lower Quartile (Q): 9
The mean, median, mode, range, variance, and Q are presented for both populations.