Following information pertains to the daily income of 150 families. Calculate the arithmetic mean. Income (in ₹ ) Number of families More than 75 150 85 140 95 115 105 95 115 70 125 60 135 40 145 25

Income Class Interval Number of Families ( c f ) Frequency ( f ) Mid Value ( m ) f m 75 - 85 150 150 - 140 = 10 80 800 85 - 95 140 140 - 115 = 25 90 2250 95 - 105 115 115 - 95 = 20 100 2000 105 - 115 95 95 - 70 = 25 110 2750 115 - 125 70 70 - 60 = 10 120 1200 125 - 135 60 60 - 40 = 20 130 2600 135 - 145 40 40 - 25 = 15 140 2100 145 - 155 25 25 150 3750 Total Σ f = 150 Σ f m = 17 , 450 Arithmetic mean is the most commonly used measure of central tendency. It is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by = Σ f m Σ f = 17450 150 = ₹ 116 . 33

Find the mean plot size for the data given, by using step deviation and assumed mean methods. Plot size in square metres Number of plots 100 200 200 50 300 10

Let assumed mean, A = 200 X f d = X - A ∑ f d 100 200 - 100 - 20000 200 50 0 0 300 10 100 1000 ∑ f = 260 ∑ f d = - 19000 Mean X ¯ = A + ∑ f d ∑ f ⇒ X ¯ = 200 + - 19000 260 = 200 - 73 . 08 = 126 . 92 In the step deviation method , along with the assumed mean, A = 200 , we divide d with a constant number c = 100 X f d = X - A d ' = X - A c ∑ f d ' 100 200 - 100 - 1 - 200 200 50 0 0 0 300 10 100 1 10 ∑ f = 260 ∑ f d ' = - 190 Mean X ¯ = A + ∑ f d ' ∑ f × c ⇒ X ¯ = 200 + - 190 260 × 100 = 200 - 73 . 08 = 126 . 92

If the arithmetic mean of the data given below is 28 , find ( a ) The missing frequency and ( b ) The median of the series. Profit per retail shop (in ₹ ) 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 Number of retail shops 12 18 27 - 17 6

( a ) Let the missing frequency be f 1 . Arithmetic Mean = 28 Profit Per Retail Shop (in ₹) Class Interval Number of Retail Shops ( f ) Mid Value ( m ) f m 0 - 10 12 5 60 10 - 20 18 15 270 20 - 30 27 25 675 30 - 40 f 1 35 35 f 1 40 - 50 17 45 765 50 - 60 6 55 330 Σ f = 80 + f 1 Σ f m = 2100 + 35 f 1 X ¯ = Σ f m Σ f or 28 = 2100 + 35 f 1 80 + f 1 or 2240 + 28 f 1 = 2100 + 35 f or 2240 - 2100 = 35 f 1 - 28 f 1 or 140 = 7 f 1 f 1 = 20 Hence, the missing frequency is 20 . ( b ) Class Interval Frequency Cumulative Frequency 0 - 10 12 12 10 - 20 18 30 20 - 30 27 57 30 - 40 20 77 40 - 50 17 94 50 - 60 6 100 Total Σ f = 100 So, the Median class = Size of N 2 t h item = 50 t h term. 50 t h item lies in the 57 t h cumulative frequency and the corresponding class interval is 20 - 30 Median = L + N 2 - C F f × i = 20 + 100 2 - 30 27 × 10 = 20 + 50 - 30 27 × 10 = 20 + 20 27 × 10 = 27 . 40

Check ∑ X - X ¯ = 0 for the following example: X : 4 , 6 , 8 , 10 , 12

Arithmetic mean is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by- X ¯ = ∑ X n = 4 + 6 + 8 + 10 + 12 5 = 40 5 = 8 ∑ X - X ¯ = ( 4 - 8 ) + ( 6 - 8 ) + ( 8 - 8 ) + ( 10 - 8 ) + ( 12 - 8 ) ∑ X - X ¯ = - 4 + ( - 2 ) + ( 0 ) + 2 + 4 = 0 Hence, proved.

Find mean and median for all four values of the series. What do you observe? Series X (Variable values) Mean Median A 1 , 2 , 3 ? ? B 1 , 2 , 30 ? ? C 1 , 2 , 300 ? ? D 1 , 2 , 3000 ? ?

Mean is average of all values whereas, median is the middle-most value. Series X (Variable values) Mean Median A 1 , 2 , 3 1 + 2 + 3 3 = 2 2 B 1 , 2 , 30 1 + 2 + 30 3 = 11 2 C 1 , 2 , 300 1 + 2 + 300 3 = 101 2 D 1 , 2 , 3000 1 + 2 + 3000 3 = 1001 2 It can be observed that mean increases but median is unaffected.

The following table gives the daily income of ten workers in a factory. Find the arithmetic mean. Workers A B C D E F G H I J Daily Income (in ₹ ) 120 150 180 200 250 300 220 350 370 260

Workers Daily Income (in ₹ ) ( X ) A 120 B 150 C 180 D 200 E 250 F 300 G 220 H 350 I 370 J 260 Total Σ X = 2400 N = 10 X ¯ = Σ X N = 2400 10 = 240 Arithmetic mean is the most commonly used measure of central tendency. It is defined as the sum of the values of all observations divided by the number of observations.

The following table gives production yield in kg. per hectare of wheat of 150 farms in a village. Calculate the mean, median and mode values. Production yield (kg. per hectare) Number of farms 50 - 53 3 53 - 56 8 56 - 59 14 59 - 62 30 62 - 65 36 65 - 68 28 68 - 71 16 71 - 74 10 74 - 77 5

Let assumed mean A = 63 . 5 Class interval Mid value d=X-A f fd Cumulative frequency 50 - 53 51 . 5 - 12 3 - 36 3 53 - 56 54 . 5 - 9 8 - 72 11 56 - 59 57 . 5 - 6 14 - 84 25 59 - 62 60 . 5 - 3 30 - 90 55 62 - 65 63 . 5 0 36 0 91 65 - 68 66 . 5 3 28 84 119 68 - 71 69 . 5 6 16 96 135 71 - 74 72 . 5 9 10 90 145 74 - 77 75 . 5 12 5 60 150 ∑ f = 150 ∑ f d = 48 Mean = A + ∑ f d ∑ f = 63 . 5 + 48 150 = 63 . 82 To find median, locate where 150 2 = 75 t h item lies. It lies in interval 62 - 65 . then median is given by formula, M = L + N 2 - C F f × h Where L is lower limit of selected class interval, f is its frequency, N is total frequency, C F is cumulative frequency of pre median class, h is class interval M = 62 + 75 - 55 36 × 3 = 63 . 67 Mode can be found by first finding modal class, which is 62 - 65 , as it is having highest frequency. Mode = L + D 1 D 1 + D 2 × h Mode = 62 + 6 6 + 8 × 3 = 63 . 28

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Arithmetic mean by following the method (assumed mean/direct) for a series of ungrouped data is calculating the sum of all observations in a series which is in turn divided by the total number of observations.

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Important Questions on Measures of Central Tendency

EASY

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

Median is unduly affected by extreme observations.

MEDIUM

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

$x : 4 6 8 10 12$

Replace the value $12$ by $96$ . What happens to the arithmetic mean. Comment.

EASY

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

The algebraic sum of deviation of a set of

n

values from

A . M .

is,

HARD

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

Which average would be suitable in this case, when the sum of absolute deviations from average is the least?

HARD

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

Following information pertains to the daily income of

150

families. Calculate the arithmetic mean.

Income (in $₹$ )	Number of families
More than $75$	$150$
$85$	$140$
$95$	$115$
$105$	$95$
$115$	$70$
$125$	$60$
$135$	$40$
$145$	$25$

MEDIUM

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

$X : 4, 6, 8, 10, 12$

In the above example, if mean is increased by $2$ , then what happens to the individual observations, if all are equally affected.

EASY

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

The sum of deviation of items from the median is zero.

HARD

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

What are the advantages of Arithmetic Mean?

MEDIUM

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

Find the mean plot size for the data given, by using step deviation and assumed mean methods.

Plot size in square metres	Number of plots
$100$	$200$
$200$	$50$
$300$	$10$

HARD

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

If the arithmetic mean of the data given below is

28,

find

(a)

The missing frequency and

(b)

The median of the series.

Profit per retail shop (in $₹$ )	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$
Number of retail shops	$12$	$18$	$27$	$-$	$17$	$6$

HARD

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

Which average would be suitable in this case, the average wage in an industrial concern?

HARD

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

Which average would be suitable in this case, average production in a factory per shift?

MEDIUM

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

Check $\sum X - \bar{X} = 0$ for the following example:

$X : 4, 6, 8, 10, 12$

MEDIUM

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

Find mean and median for all four values of the series. What do you observe?

Series	X (Variable values)	Mean	Median
$A$	$1, 2, 3$	?	?
$B$	$1, 2, 30$	?	?
$C$	$1, 2, 300$	?	?
$D$	$1, 2, 3000$	?	?

EASY

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

Which average is affected most by the presence of extreme items?

MEDIUM

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

$x : 4 6 8 10 12$

If first three items increase by $2$ , then what should be the values of the last two items, so that mean remains the same.

MEDIUM

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

Following are the runs scored by players in a match of cricket.

6

15

120

50

100

8

15

80

10

10

15

What is the mean and median?

EASY

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

Which of the following methods is used in calculating the Arithmetic Mean in grouped data in the discrete series of distribution?

HARD

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

The following table gives the daily income of ten workers in a factory. Find the arithmetic mean.

Workers	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
Daily Income (in $₹$ )	$120$	$150$	$180$	$200$	$250$	$300$	$220$	$350$	$370$	$260$

HARD

Economics>Statistics in Economics>Measures of Central Tendency>Arithmetic and Weighted Mean

The following table gives production yield in kg. per hectare of wheat of $150$ farms in a village. Calculate the mean, median and mode values.

Production yield (kg. per hectare)	Number of farms
$50 - 53$	$3$
$53 - 56$	$8$
$56 - 59$	$14$
$59 - 62$	$30$
$62 - 65$	$36$
$65 - 68$	$28$
$68 - 71$	$16$
$71 - 74$	$10$
$74 - 77$	$5$

Arithmetic mean by following the method (assumed mean/direct) for a series of ungrouped data is calculating the sum of all observations in a series which is in turn divided by the total number of observations.

Important Questions on Measures of Central Tendency

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