# Measures of Central Tendency

A measures of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. Sometimes it is are known as measures of central location . They are also classed as summary statistics. The mean (average) is most likely the measures of central tendency that you are most familiar with, but there are others, such as, the median and the mode.The mean, median and mode are all valid measures of Central Tendency but, under different conditions, some measures of central location become more appropriate to use than others. Here we will learn about

**1) Mean (Arithmetic mean) 2) Median 3) Mode.**

**Mean :**The most common central value of a group of observations is the Arithmetic mean or the Mean. If x

_{1}, x

_{2}, x

_{3}, x

_{4},…, x

_{n}are the values of n observations, then the mean of group of observations is defined as

Mean = ( sum of all observations) / Number of observations

In a discrete frequency distribution the arithmetic mean may be computed by any one of the following methods :

**1) Direct method**

2) Short- cut method

3) Step- Deviation method.

4) Merits and Demerits of Mean

2) Short- cut method

3) Step- Deviation method.

4) Merits and Demerits of Mean

__Some solved examples on Mean of Grouped Data :__1) A batsman score the following number of runs in 6 innings :

36,35,50,46,60,55. Calculate the mean runs scored in an inning.

**Solution :**Mean = ( sum of all observations) / Number of observations

Mean runs scored = ( 36 + 35 + 50 + 46 + 60 + 55) / 6 = 282 / 6 = 47

∴ Mean runs scored in an inning is 47.

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2) If the mean of five observations x, x + 2, x + 4, x + 6, x + 8 is 11. Find the mean of 1st three observations.

**Solution :**

Mean = ( sum of all observations) / Number of observations

11 = [x + ( x + 2) + (x + 4) + (x + 6) + (x + 8) ] / 5

55 = x + x + 2 + x + 4 + x + 6 + x + 8

55 = 5x + 20

5x = 35

∴ x = 7

Mean of 1st three observations,

= [ x + (x + 2) + (x + 4) ] / 3

= ( 3x + 6) / 3

= 3 (x + 2) / 3 = x +2 = 7 + 2 = 9

∴ mean of 1st three observations is 9.

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3) The mean of 40 observations was 160. It was detected on re-checking that the value 165 was wrongly copied as 125 for computation of mean. Find the correct mean.

**Solution :**

n = number of observations = 40 and Mean = 160

∴ Mean = ( sum of all observations) / Number of observations

⇒ 160 = ( sum of all observations) / 40

⇒ 160 x 40 = sum of all observations

Thus, incorrect sum of observations = 160 x 40 = 6400

Correct sum of observations = Incorrect sum – incorrect observation + correct observation

⇒ correct sum of observations = 6400 – 165 + 125

⇒ correct sum of observations = 6360

∴ correct mean = correct sum of observations / Number of observations

⇒ = 6360 / 40 = 159

**Statistics**

• Statistics

• Pictograph

• Pie chart

• Bar Graph

• Double Bar Graph

• Histogram

• Frequency polygon

• Frequency distribution (Discrete )

• Frequency distribution continuous (or grouped)

• Measures of central tendency ( Mean, Mode and Median)

• Ogive or Frequency curve.

• Statistics

• Pictograph

• Pie chart

• Bar Graph

• Double Bar Graph

• Histogram

• Frequency polygon

• Frequency distribution (Discrete )

• Frequency distribution continuous (or grouped)

• Measures of central tendency ( Mean, Mode and Median)

• Ogive or Frequency curve.