1.12. Interquartile range for grouped data

Suppose that a class, with lower boundary L and upper boundary U, contains f observations. If these observations were to be arranged in ascending order, the observation is estimated by

for

where

is the lower limit of class containing observation

is the upper limit of class containing observation

is the frequency of class containing observation

is the location of observation in that class.

For interquartile range we need to find

and

As we know .

Number of orders
10-12 13-15 16-18 19-21	4 12 20 14

Example: The following table gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a mail-order company

Calculate the interquartile range.

Solution:

First of all, let us write cumulative frequency distribution

Number of orders		Cumulative frequency
10-12 13-15 16-18 19-21	4 12 20 14	4 16 36 50

Since there are N=50 observations, we have

Hence the first quartile is the three-quarters way from the observation to . From cumulative distribution we see that the value is the value in the class 13-15. In our notation then

;

The observation is estimated by

Similarly, the observation is the value in the same class, so now, with, we have

=

Since the first quartile is three-quarters of the way from the twelves observation to the thirteens observation, we have

.

To find third quartile, we have

Therefore, when the observations are arranged in ascending order, the third quartile is half of the way from thirty-seventh to thirty-eighth.

Looking at table, we see that the thirty-seventh observation is the first value in class the 19-21, which contains t14 observations. We have then

;

Thus, the thirty-seventh observation us estimated by

Similarly, the thirty-eighth observations the second value in the same class, so with , we estimate observation by

Hence, since the third quartile is half of the way from the to ,

we have

Finally, then the interquartile range is the difference between the third and first quartiles, so

Interquartile range=

Thus, if the interquartile range is to be used as a measure of dispersion, we estimate it by.

Exercises

x	f
0 to less than 20 20 to less than 40 40 to less than 60 60 to less than 80 80 to less than 100	14 18 9 5 4

1. For 50 airplanes that arrived late at an airport during a week, the time by which they were late was observed. In the following table, x denotes the time (in minutes) by which an airplane was late and f denotes the number of airplanes.

a) Find the mean

b) Find the median

c) Find the mode

d) Find the variance and standard deviation

e) Find the interquartile range.

Amount of electric bill (dollars)	Number of families
4 to less than 8 8 to less than 12 12 to less than 16 16 to less than 20 20 to less than 24	2 9 16 8 5

2. The following table gives information on the amount (in dollars) of the electric bills for a sample of 40 families.

a) Estimate the sample mean

b) Estimate the median

c) Estimate the mode

d) Estimate the variance and standard deviation.

e) Estimate the intequartile range.

Forecast $ per share	Number of analysts
0.5-10.5 10.5-20.5 20.5-30.5 30.5-40.5	2 4 9 5

3. A population of all twenty financial analysts was asked to provide forecasts of earnings per share of a corporation for next year.

The results are summarized in the table.

a) Find the relative frequencies.

b Find the cumulative frequencies.

c Find the cumulative relative frequencies.

d) Estimate the population mean.

e) Estimate the population variance.

f) Estimate the population standard deviation.

g) Estimate the population mode.

h) Estimate the population median.

i) Estimate the intequartile range.

j) Which class is modal class?

4. A sample was taken of flights arriving at a major airport to study the problem of air traffic delays. The table shows numbers of minutes late for a sample of 100 flights.

Minutes late	0-10	10-20	20-30	30-40	40-50	50-60
Number of flights	29	23	17	14	11	6

a) Draw the histogram

b) Find the sample relative frequencies

c) Find and interpret the sample cumulative relative frequencies

d) Estimate the sample mean number of minutes

e) Estimate the sample variance and standard deviation

f) Estimate the sample median number of minutes late

g) Estimate the intequartile range

h) Which is the modal class for this sample?

Computers sold	Frequency
4 to 9 10 to 15 16 to 21 22 to 27 28 to 33	2 5 10 5 3

5. The following table gives the frequency distribution of the number of computers sold during the past 25 weeks at a computer store.

Calculate the mean, variance, and standard deviation.

Class limits	Frequency
52.5-63.5 63.5-74.5 74.5-85.5 85.5-96.5 96.5-107.5 107.5-118.5	6 12 25 18 14 5

6. Eighty randomly selected light bulbs were tested to determine their lifetimes (in hours). The following frequency distribution was obtained

Find the variance and standard deviation.

Class limits	Frequency
54-58 59-63 64-68 69-73 74-78 79-83 84-88	2 5 8 0 4 5 1

7. The following data represent the scores (in words per minute) of 25 typists on a speed test.

Find the variance and standard deviation.

8. For a sample of fifty new full-size cars, fuel consumption figures were obtained and summarized in the accompanying table

Fuel consumption	14-16	16-18	18-20	20-22	22-24
Number of cars	3	6	13	20	8

a) Draw the histogram.

b) Find the sample relative frequencies.

c) Find and interpret the sample cumulative relative frequencies

d) Estimate the sample mean fuel consumption.

e) Estimate the sample standard deviation of fuel consumption.

f) Estimate the sample median fuel consumption.

g) Estimate the sample intequartile range.

h) Which is the modal class for this sample?

9. The fuel capacity in gallons of 30 randomly selected cars is shown below.

Class	Frequency
12.5-27.5 27.5-42.5 42.5-57.5 57.5-72.5 72.5-87.5 87.5-102.5	6 3 5 8 6 2

Find

a) Mean

b) Median

c) Modal class

d) Variance

e) Standard deviation

Volts	Frequency
2 3 4 5 6	1 4 5 1 1

10. Twelve batteries were tested after being used for one hour. The output (in volts) is shown below.

Find each of the following

a) Mean

b) Median

c) Mode

d) Range

e)Variance

f) Standard deviation.

11. For a sample of twenty-five students from a large class, the accompanying table shows the amount of time students spent studying for a test

Study time (hours)	0-2	2-4	4-6	6-8	8-10
Number of students	3	4	8	7	3

a) Draw the histogram.

b) Find and interpret the cumulative relative frequencies.

c) Estimate the sample mean study time.

d) Estimate the sample median.

e) Find the modal class.

f) Estimate the sample variance.

g) Estimate the sample standard deviation study time.

h) Estimate the sample intequartile range.

Answers

1. a) ; b) 32,22; c) 30; d) ; ;

e) ; 2. a) ; b) 14,25; c) 14; d) ; ; e) ; 3. a) 2/20; 4/20; 9/20; 5/20; b) 2; 6; 15; 20;

c) 2/20; 6/20;15/20; 20/20; d) ; ;

e) ; f) median=24.944; i) I.Q.R. =13.735; j) modal class:

20.5-30.5; 4. b) 0.29; 0.23; 0.17; 0.14; 0.11; 0.06; c) 0.29; 0.52; 0.69; 0.83; 0.94; 1.0; d) ; e); ;f)

median=19.13;g) 25.931; h) modal class 0-10; 5.

6. 7. ;

8. b) 3/50; 6/50; 13/50; 20/50; 8/50; c) 3/50; 9/50; 22/50; 42/50; 50/50; d) ; e) s =2.185; f) median=20.3; g) h) modal class: 20-22; 9. a) b) median=59.4; c) modal class;

57.5-72.5; d) ;e) s =23.8; 10. a) b) median=4;

c) mode = 4; d) range = 4; e) f) s =1.05; 11. b) 3/25; 7/25; 15/25; 22/25; 25/25; c) 5.24; d) 5.375;e) modal class = 4-6; f) 5.773; g) 2.403; h) 3.64.

65