Measures of Central Tendency

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Mean

Mean: It is the central value around which the observations are scattered.

Mean = Total of all observations Total no. of observations

Mean =

2

n

xx i

Median

Median: It is that value which divides the observations into two equal parts when all observations are arranged in ascending or descending order.

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Mode

Mode: It is the observation which occurs most frequently in the data.

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Geometric MeanSometimes when we are dealing with quantities that change over a period of time, we need to know an average rate of change, such as average growth rate over a period of several years. In such cases, the simple arithmetic mean is inappropriate, because it gives a wrong answer.

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The geometric mean of a data set a1, a2, ..., an is given by

nn

nn

ii aaaa ...... 21

1

1

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E.G. A person has invested `5000 in the stock market. At the end of the first year the amount has grown to `6250; he has had a 25% profit. If at the end of the second year his principle has grown to `8750, the rate of increase is 40% for the year. What is the average rate of increase of his investment during the two years?

*If we calculate the arithmetic mean, it would be (25+40)/2 = 32.5

*Applying this rate he gets interest

= 5000(1+32.5/100)^2= `8778

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* But the actual rate of interest can be calculated as `8750 = 5000(1 + r/100)^2

On simplification gives r = 32.2876

* Now this can be explained with the concept of

geometric mean as √(1.25 x 1.40) = 1.322876

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We use the geometric mean to show multiplicative effects over time in compound interest and inflation calculations.

The geometric mean must be used when working with percentages (which are derived from values), whereas the standard arithmetic mean will work with the values themselves.

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Harmonic MeanThe harmonic mean is a better "average" when the numbers are defined in relation to some unit. The common example is averaging speed, i.e. the harmonic mean is used when averaging rates or ratios.

The harmonic mean of the positive real numbers x1, x2, ..., xn is defined to be

n

i in x

n

xxx

n

121

11....

11

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E.G. An investor buys `1200 worth of shares in a company each month. During the first five months he bought the shares at a price of `10, 12, 15, 20 and 24 per share. After 5 months what is the average price paid for the shares by him?

*If we calculate the arithmetic mean, it would be

= (10+12+15+20+24)/5 = 16.2

*The actual average price per share

= Total amount spent / Total number of shares11

Amount Spent Price per Share Number of Shares

1200 10 120

1200 12 100

1200 15 80

1200 20 60

1200 24 50

Total = 6000 Total = 410

The actual average price per share

= 6000/410 = 14.63

* Now this can be explained with the concept of harmonic mean as

63.14

241

201

151

121

101

5

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Percentile

Percentile: It is that value which divides the observations into 100 equal parts when all observations are arranged in ascending or descending order.

The kth percentile is the value which divides the data such that k% of the data lies below the value and (100-k)% of the data lies above the value.

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