Chebyshev's Theorem states that withing 2 standard deviations (regardless of the mean) that 95% of data is contained. There is also a theorem that states the minimum % of data that will lie between 2 standard deviations.
Chebyshev's Theorem states that regardless of the distribution at least (100(k^2 - 1)) / (k^2) % of the data is guarunteed to lie within k standard deviations of the mean.
a) What is the minimum % of the data that Chebyshev's Theorem guaruntees will lie within 2 standard deviations of the mean for any distribution?
b) Verify the prediction in a) for the data 0, 9, 9.
To find the minimum % of data that Chebyshev's Theorem guarantees will lie within 2 standard deviations of the mean for any distribution, we can use the formula provided: (100(k^2 - 1)) / (k^2) %.
a) For this specific case, we need to substitute k = 2 into the formula. Plugging in the values, we have:
(100(2^2 - 1)) / (2^2) %
(100(4 - 1)) / 4 %
(100 * 3) / 4 %
300 / 4 %
75%
Therefore, Chebyshev's Theorem guarantees that at least 75% of the data will lie within 2 standard deviations of the mean for any distribution.
b) To verify the prediction for the data set 0, 9, 9, we need to calculate the mean and the standard deviation. Then we can determine the percentage of data that falls within 2 standard deviations from the mean.
- Step 1: Calculate the mean
The mean is obtained by summing up all the values and dividing by the number of values.
Mean = (0 + 9 + 9) / 3
Mean = 18 / 3
Mean = 6
- Step 2: Calculate the standard deviation
The standard deviation measures the spread of data around the mean. We can use the formula for sample standard deviation:
Standard Deviation = sqrt(((0 - 6)^2 + (9 - 6)^2 + (9 - 6)^2) / (3 - 1))
Calculating each component:
(0 - 6)^2 = 36
(9 - 6)^2 = 9
(9 - 6)^2 = 9
Adding them up:
36 + 9 + 9 = 54
Now, divide by (3 - 1):
54 / 2 = 27
Taking the square root:
sqrt(27) ≈ 5.20
Therefore, the standard deviation is approximately 5.20.
- Step 3: Determine the percentage within 2 standard deviations
To find the percentage of data that falls within 2 standard deviations from the mean, we need to check how many data points lie within the range of Mean ± (2 * Standard Deviation).
Mean ± (2 * Standard Deviation) = 6 ± (2 * 5.20)
Mean ± (2 * Standard Deviation) = 6 ± 10.40
This means the range is from -4.40 to 16.40.
Out of the three data points (0, 9, 9), only 2 fall within this range, which are 0 and 9. Therefore, the percentage of data that lies within 2 standard deviations is 2 / 3 * 100% ≈ 66.67%.
Comparing the result with the predicted minimum % from Chebyshev's Theorem (75%), we can see that the prediction does not hold for this specific data set in terms of the minimum % guarantee.