General Statistics
Examples
Descriptive Statistics - Introduction


Click on Graphs and images to see larger picture  (Programs Used: GraphsBasic Statistics, Group Statistics )
Question 1 - A podiatrist recorded the recovery times, measured in days, for 36 patients.
He was trying out new procedure and hoped that the recovery times would be less than the usual 6 days.
The recovered times are:
 
8 7 6 9 4 5 3 7 8
10 7 7 6 4 10 3 6 8
2 5 4 5 3 8 7 4 6
3 7 12 4 3 6 6 9 4

(a) Construct the frequency distribution.

(b) What percentage of recovery times were less than 10 days?

Solutions:

(a) The frequency distribution (the number of times each occurred) of each recovery times (range from 2 to 12) is:
 
Recovery Times, x 2 3 4 5 6 7 8 9 10 12 Total
Frequency, f 1 5 6 3 6 6 4 2 2 1 36

(b) Sum of frequencies up to and including 9 is 33 so percentage less than 10 is 

Question 2. The owner of a small business wants to analyze the profits (thousands of dollars) over the past 30 years.
The ordered data from smallest to largest profits are as follows:
 
15 17 18 19 20 20 20 21 23 23 24 24 24 24 24
25 25 25 25 25 26 26 27 27 28 29 30 30 31 32

(a) Complete the grouped frequency distribution table:
 
Class Class boundaries Frequency
     
2 17.5-20.5  
     

(b) What is the class width?
(c) Compute the relative frequency of each class
(d) What is the median and mode?

Solutions:

(a), (b) and (c) Relative Frequency table - Class width is 20.5 - 17.5 = 3
 
Class Class boundaries Frequency

(1)

Relative Frequency

(1) / (2)

1 14.5-17.5 2 0.667
2 17.5-20.5 5 0.167
3 20.5-23.5 3 0.1
4 23.5-26.5 12 0.4
5 26.5-29.5 4 0.133
6 29.5-32.5 4 0.133
Total
  (2) Sum = 30 Sum =1

(d) The mode is 24 and 25 since they appears the most, 5 times

The median is (24 + 25) /2 = 24.5

Question 3 - The following is the world distribution of nuclear reactors in operation:
 
United States 109 Russia 29
France 56 Canada 21
Japan 51 Germany 20
United Kingdom 35 Others 116
(a) Construct a bar graph

(c) Construct a pie chart

Question 4. From the following data (per 1000 population) for selected countries:
Compute the statistics required for a box-plot diagram: Ordered Data
 
4.9 5.1 6.3 6.7 7 7.1 7.8 8.6 8.7 8.8  
8.9 9 9.4 9.4 9.5 9.6 9.9 10 10.1 10.1  
10.1 10.3 10.4 10.7 10.9 11.3 11.6 11.8 14.3    

Solutions: (see programs)
 
min 25 percentile median 75 percentile max
4.9 8.6 9.5 10.3 14.3
Basic Statistics - Mean and Variance        
               
Enter Data         Basic Statistical Parameters
4.9 5.1 6.3 6.7 7 Sample,n 29  
8.9 9 9.4 9.4 9.5 mean 9.251724138  
10.1 10.3 10.4 10.7 10.9 std dev, s 2.040766783  
7.1 7.8 8.6 8.7 8.8 variance 4.164729064  
9.6 9.9 10 10.1 10.1 median 9.5  
11.3 11.6 11.8 14.3   mode 10.1 Look for more
          range 9.4  
          max 14.3  
          min 4.9  
               
Percentile     Box Plot   Inter quartile Range = 1.7
100% 14.3   Min 4.9 Semi-inter quartile range = 0.85
75% 10.3   25% Pt 8.6      
50% 9.5   Median 9.5    
25% 8.6   75% Pt 10.3      
10% 6.62   Max 14.3      
               

Question 5 The following grouped frequency distribution represents the ages (in years) of 59 patients of a counseling center.

(a) Compute a frequency polygon and

(b) Compute the mean and standard deviation of the age of the patients.
 
Class Limits 21-27 28-34 35-41 42-48 49-55 56-62 63-69
Frequency 3 7 12 15 12 7 3

Solution.
 
Class Limits 21-27 28-34 35-41 42-48 49-55 56-62 63-69
Class Midpoint 24 31 38 45 52 59 66
Frequency 3 7 12 15 12 7 3
(a) Frequency polygon

(b) Weighted Mean and Grouped Standard Deviations

Weighted Mean = 45

Group variance = 113.2069

Group Standard Deviation = 10.64

Program Output

Question 6. The following data gives the average temperatures per month over a
12-month period for two cities A and B.
 
City A 8 14 25 43 54 64 71 69 58 47 29 16
City B 56 60 58 62 63 68 69 71 69 67 61 58

(a) Which of the two cities do you suspect is warmer on average? Which has more temperature variability?

(b) Use the following information and compute the mean and standard deviation for each city.

City A  and 

City B  and 
 
(a) City B has the larger mean of 63.5

And City A has a larger variation in temperature with a larger variance of 504.64

(b) From Formula:

City A  and

City B  and

City A: 

= 22.46 mean = 

City B: mean = 63.5 and s=5.11

Variance = 26.09

Example of calculation for City B

Question 6 The following data give the time in days from remission to relapse for 51 patients with an acute illness.
 

Find the following:
 
Questions Solutions from Program
(a) The first and third quartiles 135.5 and 367
(b) The inter quartile range 231.5
(c) The median 249
(d) The 95th percentile 669.5
(e) The percentile rank of 111 20%