n = the total frequency F = the cumulative frequency before class median i = the class width = the lower boundary of the class median
Suppose that we want to find the median height of the class of the first school children This is the cumulative frequency distribution Example 4 Height (cm)
Cumulative frequencies are useful if more detailed information is required about a set of data In particular, they can be used to find the median and
Cumulative frequency of each class is the sum of the frequency of the class and the frequencies of the pervious classes, ie adding the frequencies successively,
The “cumulative frequency” is the sum of the frequencies of that class and all previous classes Example Add the midpoint of each class, the relative frequency
e) Determine the modal class, median class and mean (to the nearest whole number) f) Sketch a frequency histogram and polygon g) Sketch a cumulative
Represent cumulative frequency, draw histogram, frequency polygon where L = Lower limit of median class; m = Cumulative frequency above median class
Read each question carefully before you begin answering it (b) Use the cumulative frequency diagram to estimate the median
The mean, median and mode are all valid measures of central tendency, So Median = 12 (since total in the cumulative frequency column which is equal to
and statistics using a frequency distribution Don't forget frequency times the class midpoint cumulative frequency before the median's frequency
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69464_3GroupedDataCalculation.pdf 1.
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.