[PDF] Lecture 2 – Grouped Data Calculation

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Mean, Median and Mode

2.

First Quantile, third Quantile

and

Interquantile

Range.

Lecture 2 - Grouped Data

Calculation

Mean - Grouped Data

Number

of orderf

10 - 12

13 - 15

16 - 18

19 - 214

12 20 14 n = 50

Number

of orderfxfx

10 - 12

13 - 15

16 - 18

19 - 214

12 20

141114

17 2044
168
340
280
n = 50= 832 fx832 x = = =16.64 n50 Example: The following table gives the frequency distribution of the nu mber of orders received each day during the past 50 days at the office of a m ail-order company. Calculate the mean.

Solution:

X is the midpoint of the

class. It is adding the class limits and divide by 2.

Median and Interquartile Range -Grouped Data

Step 1:

Construct the cumulative frequency distribution.

Step 2:

Decide the class that contain the median.

Class Median

is the first class with the value of cumulative frequency equal at least n/2.

Step 3:

Find the median by using the following formula:

Median

m m n-F2=L +if m L m f

Where:

n = the total frequency F = the cumulative frequency before class median i = the class width = the lower boundary of the class median= the frequency of the class median

Time to travel to workFrequency

1 - 10

11 - 20

21 - 30

31 - 40

41 - 508

14 12 9 7 Example: Based on the grouped data below, find the median:

Solution:

Time to travel

to workFrequencyCumulative

Frequency

1 - 10

11 - 20

21 - 30

31 - 40

41 - 508

14 12 9 78
22
34
43
50
25250
2n m f m L 1 st Step: Construct the cumulative frequency distribution class median is the 3 rd class

So,

F = 22, = 12, = 20.5 and i = 10

Therefore,

2 25 22

2151012

24

Median

= = m m n-F =L if - . Thus, 25 persons take less than 24 minutes to travel to work and another 25 persons take more than 24 minutes to travel to work. 1 1 1Q Q n-F4QL+ if 3 3 3Q Q

3n-F4QL+ if

QuartilesUsing the same method of calculation as in the Median, we can get Q 1 and Q 3 equation as follows:

Time to travel to workFrequency

1 - 10

11 - 20

21 - 30

31 - 40

41 - 508

14 12 9 7 Example: Based on the grouped data below, find the Interquartile Range

Time to travel

to workFrequencyCumulative

Frequency

1 - 10

11 - 20

21 - 30

31 - 40

41 - 508

14 12 9 78
22
34
43
50
1 n50Class Q12 544. 1 1 1 4 125 8

105 1014

137143

Q Q n-F QL if .- . .

Solution:

1 st Step: Construct the cumulative frequency distribution

Class Q

1 is the 2 nd class

Therefore,2

nd

Step: Determine the Q

1 and Q 3 3

3503nClass Q37 544.

3 3 3 4

375 34

305109

343889

Q Q n-F QL if .- . .

IQR = Q

3 -Q 1

Class Q

3 is the 4 th class

Therefore,

Interquartile Range

IQR = Q

3 -Q 1 calculate the IQ

IQR = Q

3 -Q 1 = 34.3889 - 13.7143 = 20.6746 Mode•Mode is the value that has the highest frequency in a data set. •For grouped data, class mode (or, modal class) is the class with the h ighest frequency. •To find mode for grouped data, use the following formula: Mode 1 mo 12

ǻ=L +iǻ+ǻ

Mode - Grouped Data

mo L 1 2

Where:

is the lower boundary of class modeis the difference between the frequency of class mode and the frequency of the class before the class modeis the difference between the frequency of class mode and the frequency of the class after the class modei is the class width

Calculation of Grouped Data - Mode

Time to travel to workFrequency

1 - 10

11 - 20

21 - 30

31 - 40

41 - 508

14 12 9 7 Example: Based on the grouped data below, find the mode mo L 1 2

610 510 17 562Mode=..

Solution: Based on the table,

= 10.5, = (14 - 8) = 6, = (14 - 12) = 2 and i = 10

Mode can also be obtained from a histogram.Step 1: Identify the modal class and the bar representing it

Step 2: Draw two cross lines as shown in the diagram. Step 3: Drop a perpendicular from the intersection of the two lines until it touch the horizontal axis.

Step 4: Read the mode from the horizontal axis

2 2 2 fxfxN N 2 2 2 1 fxfxnsn 22
22
ss

Population Variance:

Variance for sample data:

Standard Deviation:

Population:

Sample:

Variance and Standard Deviation -Grouped Data

No. of orderf

10 - 12

13 - 15

16 - 18

19 - 214

12 20 14

Totaln = 50

No. of orderfxfxfx

2

10 - 12

13 - 15

16 - 18

19 - 214

12 20 1411
14 17 2044
168
340

280484

2352
5780
5600

Totaln = 5083214216

Example: Find the variance and standard deviation for the following data:

Solution:

2 2 2 2 1

8321421650

50 1
75820
fx fxnsn .

75.25820.7

2 ss

Variance,

Standard Deviation,

Thus, the standard deviation of the number of orders received at the office of this mail-order company during the past 50 days is 2.75.

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