Mean, Median, Five Number Summary

1. Consider a sample with data values of 27, 25, 20, 15, 30, 34, 28, and 25. a) Compute the mean, median, and mode.
b) Compute the 20th, 65th, and 75th percentiles.
c) Compute the range, interquartile range, variance, and standard deviation.

Answers:
Data values: 15, 20, 25, 25, 27, 28, 30, 34
a) Mean: [pic]= ∑xi/n = (15+20+25+25+27+28+30+34)/8 = 204/8 = 25.5 Median: Even number, so median is = (25+27)/2 = 26
Mode: Most frequent number = 25

b) 20th Percentile = (P/100)n = (20/100)8 = 0.2×8 = 1.6 = 2 65th Percentile = (65/100)8 = 0.65×8 = 5.2 = 6
75th Percentile = (75/100)8 = 0.75×8 = 6

c) Range: Largest data value – smallest data value = 34-15 = 19 Interquartile range: 3rd Quartile – 1st Quartile = 6 –((25/100)8) = 6-2 = 4 Variance: [pic]= (204-25.5)/8-1 = 25.5
Standard Deviation: [pic]= [pic]= 5.05

2. Consider a sample with a mean of 500 and a standard deviation of 100. What are the z-scores for the following data values: 650, 500, and 280?

Answers:
a) Data value 650: z-scores = [pic] = (650-500)/100 = 1.5 b) Data value 500: (500-500)/100 = 0
c) Data value 280: (280-500)/100 = -2.2

3. Consider a sample with a mean of 30 and a standard deviation of 5. Use Chebyshev’s theorem to determine the percentage of the data within each of the following ranges. a) 20 to 40
b) 15 to 45
c) 18 to 42

Answers:
Standard deviation (s) = 5
Mean ([pic]) = 30
a) z1 = (20-30)/5 = -2
z2 = (40-30)/5 = 2
(1-1/z²) = (1-1/2²) = 0.75 = 75%

b) z1 = (15-30)/5 = -3
z2 = (45-30)/5 = 3
(1-1/3²) = 0.89 = 89%

c) z1 = (18-30)/5 = -2.4
z2 = (42-30)/5 = 2.4
(1-1/2.4²) = 0.83 = 83%

4. Suppose the data have a bell-shaped distribution with a mean of 30 and a standard deviation of 5. Use the empirical rule to determine the percentage of data within each of the following ranges. a) 20 to 40

b) 15 to 45
c) 25 to 35

Answers:
Standard deviation (s) = 5
Mean ([pic]) = 30
a) z1 = (20-30)/5 = -2
z2 = (40-30)/5 = 2
By empirical rule, there are at approximately 95% of the data values will be within this interval.

b) z1 = (15-30)/5 = -3
z2 = (45-30)/5 = 3
By empirical rule, there are at approximately 99.7% of the data values will be within this interval.

c) z1 = (25-30)/5 = -1
z2 = (35-30)/5 = 1
By empirical rule, there are at approximately 68% of the data values will be within this interval.