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This pdf includes the following topics:-

Mean – Grouped Data

Median and Interquartile Range – Grouped Data

Quartiles

Mode – Grouped Data

Variance and Standard Deviation-Grouped Data

Mean – Grouped Data

Median and Interquartile Range – Grouped Data

Quartiles

Mode – Grouped Data

Variance and Standard Deviation-Grouped Data

1.
Lecture 2 – Grouped Data

Calculation

1. Mean, Median and Mode

2. First Quantile, third Quantile and

Interquantile Range.

Calculation

1. Mean, Median and Mode

2. First Quantile, third Quantile and

Interquantile Range.

2.
Mean – Grouped Data

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 mean. Number f

of order

10 – 12 4

13 – 15 12

16 – 18 20

19 – 21 14

n = 50

X is the midpoint of the

Number f x fx class. It is adding the class

of order limits and divide by 2.

10 – 12 4 11 44

13 – 15 12 14 168

x=

∑ fx = 8 3 2 = 1 6 .6 4

16 – 18 20 17 340 n 50

19 – 21 14 20 280

n = 50 = 832

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 mean. Number f

of order

10 – 12 4

13 – 15 12

16 – 18 20

19 – 21 14

n = 50

X is the midpoint of the

Number f x fx class. It is adding the class

of order limits and divide by 2.

10 – 12 4 11 44

13 – 15 12 14 168

x=

∑ fx = 8 3 2 = 1 6 .6 4

16 – 18 20 17 340 n 50

19 – 21 14 20 280

n = 50 = 832

3.
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:

⎛ n ⎞

⎜ 2 - F ⎟

M e d ia n = L m + ⎜ ⎟ i

⎜ f m ⎟

⎝ ⎠

Where:

n = the total frequency

F = the cumulative frequency before class median

f = the frequency of the class median

m

i = the class width

Lm = the lower boundary of the class median

– 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:

⎛ n ⎞

⎜ 2 - F ⎟

M e d ia n = L m + ⎜ ⎟ i

⎜ f m ⎟

⎝ ⎠

Where:

n = the total frequency

F = the cumulative frequency before class median

f = the frequency of the class median

m

i = the class width

Lm = the lower boundary of the class median

4.
Example: Based on the grouped data below, find the median:

Time to travel to work Frequency

1 – 10 8

11 – 20 14

21 – 30 12

31 – 40 9

41 – 50 7

1st Step: Construct the cumulative frequency distribution

Time to travel Frequency Cumulative

to work Frequency

1 – 10 8 8

11 – 20 14 22

21 – 30 12 34

31 – 40 9 43

41 – 50 7 50

n 50

= = 25 class median is the 3rd class

2 2

So, F = 22, fm = 12, L = 20.5 and i = 10

m

Time to travel to work Frequency

1 – 10 8

11 – 20 14

21 – 30 12

31 – 40 9

41 – 50 7

1st Step: Construct the cumulative frequency distribution

Time to travel Frequency Cumulative

to work Frequency

1 – 10 8 8

11 – 20 14 22

21 – 30 12 34

31 – 40 9 43

41 – 50 7 50

n 50

= = 25 class median is the 3rd class

2 2

So, F = 22, fm = 12, L = 20.5 and i = 10

m

5.
⎛n ⎞

⎜ - F ⎟

Median = Lm + ⎜ 2 ⎟i

f

⎜ m ⎟

⎝ ⎠

⎛ 25 - 22 ⎞

= 21.5 + ⎜ ⎟ 10

⎝ 12 ⎠

= 24

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.

⎜ - F ⎟

Median = Lm + ⎜ 2 ⎟i

f

⎜ m ⎟

⎝ ⎠

⎛ 25 - 22 ⎞

= 21.5 + ⎜ ⎟ 10

⎝ 12 ⎠

= 24

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.

6.
Using the same method of calculation as in the Median,

we can get Q1 and Q3 equation as follows:

⎛n ⎞ ⎛ 3n ⎞

⎜4-F ⎟ ⎜ 4 -F ⎟

Q1 = LQ1 + ⎜ ⎟i Q3 = LQ3 + ⎜ ⎟i

⎜ f Q1 ⎟ ⎜ f Q3 ⎟

⎝ ⎠ ⎝ ⎠

Example: Based on the grouped data below, find the Interquartile Range

Time to travel to work Frequency

1 – 10 8

11 – 20 14

21 – 30 12

31 – 40 9

41 – 50 7

we can get Q1 and Q3 equation as follows:

⎛n ⎞ ⎛ 3n ⎞

⎜4-F ⎟ ⎜ 4 -F ⎟

Q1 = LQ1 + ⎜ ⎟i Q3 = LQ3 + ⎜ ⎟i

⎜ f Q1 ⎟ ⎜ f Q3 ⎟

⎝ ⎠ ⎝ ⎠

Example: Based on the grouped data below, find the Interquartile Range

Time to travel to work Frequency

1 – 10 8

11 – 20 14

21 – 30 12

31 – 40 9

41 – 50 7

7.
1st Step: Construct the cumulative frequency distribution

Time to travel Frequency Cumulative

to work Frequency

1 – 10 8 8

11 – 20 14 22

21 – 30 12 34

31 – 40 9 43

41 – 50 7 50

2nd Step: Determine the Q1 and Q3

n 50 ⎛n ⎞

Class Q 1 = = = 12 . 5

4 4 ⎜ 4 -F ⎟

Q1 = LQ1 + ⎜ ⎟i

⎜ fQ1

Class Q1 is the 2nd class

⎟

Therefore, ⎝ ⎠

⎛ 12.5 - 8 ⎞

= 10.5 + ⎜ ⎟ 10

⎝ 14 ⎠

= 13.7143

Time to travel Frequency Cumulative

to work Frequency

1 – 10 8 8

11 – 20 14 22

21 – 30 12 34

31 – 40 9 43

41 – 50 7 50

2nd Step: Determine the Q1 and Q3

n 50 ⎛n ⎞

Class Q 1 = = = 12 . 5

4 4 ⎜ 4 -F ⎟

Q1 = LQ1 + ⎜ ⎟i

⎜ fQ1

Class Q1 is the 2nd class

⎟

Therefore, ⎝ ⎠

⎛ 12.5 - 8 ⎞

= 10.5 + ⎜ ⎟ 10

⎝ 14 ⎠

= 13.7143

8.
⎛n ⎞

3n 3 ( 50 ) ⎜ 4 -F ⎟

Class Q 3 = = = 37 .5 Q3 = LQ3 + ⎜ ⎟i

⎜ fQ3

4 4 ⎟

⎝ ⎠

⎛ 37.5 - 34 ⎞

Class Q3 is the 4th class = 30.5 + ⎜ ⎟ 10

Therefore, ⎝ 9 ⎠

= 34.3889

Interquartile Range

IQR = Q3 – Q1

IQR = Q3 – Q1

calculate the IQ

IQR = Q3 – Q1 = 34.3889 – 13.7143 = 20.6746

3n 3 ( 50 ) ⎜ 4 -F ⎟

Class Q 3 = = = 37 .5 Q3 = LQ3 + ⎜ ⎟i

⎜ fQ3

4 4 ⎟

⎝ ⎠

⎛ 37.5 - 34 ⎞

Class Q3 is the 4th class = 30.5 + ⎜ ⎟ 10

Therefore, ⎝ 9 ⎠

= 34.3889

Interquartile Range

IQR = Q3 – Q1

IQR = Q3 – Q1

calculate the IQ

IQR = Q3 – Q1 = 34.3889 – 13.7143 = 20.6746

9.
Mode – Grouped Data

•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 highest frequency.

•To find mode for grouped data, use the following formula:

⎛ Δ1 ⎞

M o d e = Lmo + ⎜ ⎟i

Δ

⎝ 1 + Δ 2 ⎠

i is the class width

Δ1 is the difference between the frequency of class mode and the frequency

of the class after the class mode

Δ 2 is the difference between the frequency of class mode

and the frequency of the class before the class mode

Lmo is the lower boundary of class 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 highest frequency.

•To find mode for grouped data, use the following formula:

⎛ Δ1 ⎞

M o d e = Lmo + ⎜ ⎟i

Δ

⎝ 1 + Δ 2 ⎠

i is the class width

Δ1 is the difference between the frequency of class mode and the frequency

of the class after the class mode

Δ 2 is the difference between the frequency of class mode

and the frequency of the class before the class mode

Lmo is the lower boundary of class mode

10.
Calculation of Grouped Data - Mode

Example: Based on the grouped data below, find the mode

Time to travel to work Frequency

1 – 10 8

11 – 20 14

21 – 30 12

31 – 40 9

41 – 50 7

Based on the table,

Lmo = 10.5, Δ1 = (14 – 8) = 6, Δ 2 = (14 – 12) = 2 and

i = 10

⎛ 6 ⎞

M o d e = 1 0 .5 + ⎜ ⎟ 1 0 = 1 7 .5

⎝ 6 + 2 ⎠

Example: Based on the grouped data below, find the mode

Time to travel to work Frequency

1 – 10 8

11 – 20 14

21 – 30 12

31 – 40 9

41 – 50 7

Based on the table,

Lmo = 10.5, Δ1 = (14 – 8) = 6, Δ 2 = (14 – 12) = 2 and

i = 10

⎛ 6 ⎞

M o d e = 1 0 .5 + ⎜ ⎟ 1 0 = 1 7 .5

⎝ 6 + 2 ⎠

11.
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

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

12.
Variance and Standard Deviation

-Grouped Data

( ∑ fx )

2

Population Variance: ∑ fx 2

−

N

σ2 =

N

( ∑ fx )

2

∑ fx 2

−

n

Variance for sample data: s =

2

n −1

Standard Deviation:

Population: σ2 = σ2

Sample: s2 = s2

-Grouped Data

( ∑ fx )

2

Population Variance: ∑ fx 2

−

N

σ2 =

N

( ∑ fx )

2

∑ fx 2

−

n

Variance for sample data: s =

2

n −1

Standard Deviation:

Population: σ2 = σ2

Sample: s2 = s2

13.
Example: Find the variance and standard deviation for the following data:

No. of order f

10 – 12 4

13 – 15 12

16 – 18 20

19 – 21 14

Total n = 50

Solution:

No. of order f x fx fx2

10 – 12 4 11 44 484

13 – 15 12 14 168 2352

16 – 18 20 17 340 5780

19 – 21 14 20 280 5600

Total n = 50 832 14216

No. of order f

10 – 12 4

13 – 15 12

16 – 18 20

19 – 21 14

Total n = 50

Solution:

No. of order f x fx fx2

10 – 12 4 11 44 484

13 – 15 12 14 168 2352

16 – 18 20 17 340 5780

19 – 21 14 20 280 5600

Total n = 50 832 14216

14.
( ∑ fx )

2

∑ fx 2

−

n

Variance, s2 =

n −1

(832 )

2

14216 −

= 50

50 − 1

= 7.5820

Standard Deviation, s = s = 7.5820 = 2.75

2

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.

2

∑ fx 2

−

n

Variance, s2 =

n −1

(832 )

2

14216 −

= 50

50 − 1

= 7.5820

Standard Deviation, s = s = 7.5820 = 2.75

2

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.