Exercise 3: 1. Recall the Chebyshev' rule. 2. Given the following data sets. The sample mean and sample standard deviation are as follows: a. Which data set has a larger variation? (justify your answer) b. Is Chebyshev' rule verified for these two data sets. (Justify your answer graphically). c. What can we conclude? 3. In 1903, K. Pearson and A. Lee published the paper “On the Laws of Inheritance in Man. I. Inheritance of Physical Characters” (Biometrika, Vol. 2, pp. 357–462). The article examined data on forearm length, in inches, for a sample of 140 men. The mean and standard deviation of the forearm lengths are 18.8 in. and 1.12 in., respectively. a. Apply Chebyshev’s rule with k = 2 to make pertinent statements about the forearm lengths of the men in the sample. b. Repeat part (a) with k = 3.
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Exercise 3:
1. Recall the Chebyshev' rule.
2. Given the following data sets.
Data Set I
41
44
45
47
47
48
51
53
58
66
Data Set II
20
37
48
48
49
50
53
61
64
70
The sample mean and sample standard deviation are as follows:
Data Set I
Data Set II
i = 50.0
i = 50.0
s = 14.2
s = 7,4
a. Which data set has a larger variation? (justify your answer)
b. Is Chebyshev' rule verified for these two data sets. (Justify your answer graphically).
c. What can we conclude?
3. In 1903, K. Pearson and A. Lee published the paper "On the Laws of Inheritance in Man. I.
Inheritance of Physical Characters" (Biometrika, Vol. 2, pp. 357-462). The article examined data on
forearm length, in inches, for a sample of 140 men. The mean and standard deviation of the forearm
lengths are 18.8 in. and 1.12 in., respectively.
a. Apply Chebyshev's rule with k = 2 to make pertinent statements about the forearm lengths
of the men in the sample.
b. Repeat part (a) with k= 3.
C3
oull lia SI p
ENG
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
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