Measures of Central Tendency: Mean, Median, and Mode: Formula,explanation and examples.

ARITHMETIC MEAN Suppose the monthly income (in Rs) of six families is given as: 1600, 1500, 1400, 1525, 1625, 1630. The mean family income is obtained by adding up the incomes and dividing by the number of families.
= 1600+1500+1400+1525+1625+1630
6
= Rs 1,547
It implies that on an average, a family earns Rs 1,547. Arithmetic mean is the most commonly used measure of central tendency.

How Arithmetic Mean is Calculated The calculation of arithmetic mean can be studied under two broad categories:
1. Arithmetic Mean for Ungrouped Data.
2. Arithmetic Mean for Grouped Data. Measures of Central Tendency: Mean, Median, and Mode: Formula,explanation and examples.

Arithmetic Mean for Series of Ungrouped Data
Direct Method
Arithmetic mean by direct method is the sum of all observations in a series divided by the total number of observations.

Example 1: Calculate Arithmetic Mean from the data showing marks of students in a class in an economics test: 40, 50, 55, 78, 58.

X= ΣX /N =
Σ = 40 +50+ 55+ 78+ 58/5

= 56.2

Step Deviation Method
The calculations can be further simplified by dividing all the deviations taken from assumed mean by the common factor ‘c’.

Assumed Mean Method: If the number of observations in the data is more and/or figures are large, it is difficult to compute arithmetic mean by direct method. The computation can be made easier by using assumed mean method. In order to save time in calculating mean from a data set containing a large number of observations as well as large numerical figures, you can use assumed mean method. Here you assume a particular figure in the data as the arithmetic mean on the basis of logic/experience. Then you may take deviations of the said assumed mean from each of the observation. You can, then, take the summation of these deviations and divide it by the number of observations in the data. The actual arithmetic mean is estimated by taking the sum of the assumed mean and the ratio of sum of deviations to number of observations. Symbolically.

Continuous Series: Here, class intervals are given. The process of calculating arithmetic mean in case of continuous series is same as that of a discrete series. The only difference is that the mid-points of various class intervals are taken. We have already known that class intervals may be exclusive or inclusive or of unequal size. Example of exclusive class interval is, say, 0–10, 10–20 and so on. Example of inclusive class interval is, say, 0–9, 10–19 and so on. Example of unequal class interval is, say, 0–20, 20–50 and so on. In all these cases, calculation of arithmetic mean is done in a similar way.

Discrete Series :In case of discrete series the position of median i.e. (N+1)/2th item can be located through cumulative frequency. The corresponding value at this position is the value of median.

Weighted Arithmetic Mean :Sometimes it is important to assign weights to various items according to their importance when you calculate the arithmetic mean. For example, there are two commodities, mangoes and potatoes. You are interested in finding the average price of mangoes (P1 ) and potatoes (P2 ). The arithmetic mean will be P1+P2/2. However, you might want to give more importance to the rise in price of potatoes (P2 ). To do this, you may use as ‘weights’ the share of mangoes in the budget of the consumer (W1 ) and the share of potatoes in the budget (W2 ). Now the arithmetic mean weighted by the shares in the budget would be
= W1 P1 + W2 P2  .
W1+W2

MEDIAN: Median is that positional value of the variable which divides the distribution into two equal parts, one part comprises all values greater than or equal to the median value and the other comprises all values less than or equal to it. The Median is the “middle” element when the data set is arranged in order of the magnitude. Since the median is determined by the position of different values, it remains unaffected if, say, the size of the largest value increases.

Computation of median :The median can be easily computed by sorting the data from smallest to largest and finding out the middle value.

Example: Suppose we have the following observation in a data set: 5, 7, 6, 1, 8, 10, 12, 4, and 3. Arranging the data, in ascending order you have: 1, 3, 4, 5, 6, 7, 8, 10, 12.

=The “middle score” is 6, so the median is 6. Half of the scores are larger than 6 and half of the scores are smaller. If there are even numbers in the data, there will be two observations which fall in the middle. The median in this case is computed as the arithmetic mean of the two middle values.
=  6

MODE: Sometimes, you may be interested in knowing the most typical value of a series or the value around which maximum concentration of items occurs. For example, a manufacturer would like to know the size of shoes that has maximum demand or style of the shirt that is more frequently demanded. Here, Mode is the most appropriate measure. The word mode has been derived from the French word “la Mode” which signifies the most fashionable values of a distribution, because it is repeated the highest number of times in the series. Mode is the most frequently observed data value. It is denoted by Mo.
Computation of Mode
Discrete Series
Consider the data set 1, 2, 3, 4, 4, 5. The mode for this data is 4 because 4 occurs most frequently (twice) in the data.

Quartiles Quartiles are the measures which divide the data into four equal parts, each portion contains equal number of observations. There are three quartiles. The first Quartile (denoted by Q1 ) or lower quartile has 25% of the items of the distribution below it and 75% of the items are greater than it. The second Quartile (denoted by Q2 ) or median has 50% of items below it and 50% of the observations above it. The third Quartile (denoted by Q3 ) or upper Quartile has 75% of the items of the distribution below it and 25% of the items above it. Thus, Q1 and Q3 denote the two limits within which central 50% of the data lies.
Calculation of Quartiles The method for locating the Quartile is same as that of the median in case of individual and discrete series. The value of Q1 and Q3 of an ordered series can be obtained by the following.
formula where N is the number of observations.
Q1 = size of (N + 1) 4 th item
Q3 = size of 3(N+1) 4 th item.

Percentiles Percentiles divide the distribution into hundred equal parts, so you can get 99 dividing positions denoted by P1 , P2 , P3 , ..., P99. P50 is the median value. If you have secured 82 percentile in a management entrance examination, it means that your position is below 18 per cent of total candidates appeared in the examination. If a total of one lakh students appeared, where do you stand?

CONCLUSION
Measures of central tendency or averages are used to summarise the data. It specifies a single most representative value to describe the data set. Arithmetic mean is the most commonly used average. It is simple to calculate and is based on all the observations. But it is unduly affected by the presence of extreme items. Median is a better summary for such data. Mode is generally used to describe the qualitative data. Median and mode can be easily computed graphically. In case of open-ended distribution they can also be easily computed. Thus, it is important to select an appropriate average depending upon the purpose of analysis and the nature of the distribution

Golden rules to follow to calculate for the measurement of  Central Tendency.

• The measure of central tendency summarises the data with a single value, which can represent the entire data.
• Arithmetic mean is defined as the sum of the values of all observations divided by the number of observations.
• The sum of deviations of items from the arithmetic mean is always equal to zero.
• Sometimes, it is important to assign weights to various items according to their importance.
• Median is the central value of the distribution in the sense that the number of values less than the median is equal to the number greater than the median.
• Quartiles divide the total set of values into four equal parts.
• Mode is the value which occurs most frequently.

EXERCISES

1. Which average would be suitable in the following cases?
(ii) Average intelligence of students in a class.
(iii) Average production in a factory per shift.
(iv) Average wage in an industrial concern.
(v) When the sum of absolute deviations from average is least.
(vi) When quantities of the variable are in ratios.
(vii)In case of open-ended frequency distribution.

2. Indicate the most appropriate alternative from the multiple choices provided against each question.

(i) The most suitable average for qualitative measurement is
(a) arithmetic mean
(b) median
(c) mode
(d) geometric mean
(e) none of the above

(ii) Which average is affected most by the presence of extreme items?
(a) median
(b) mode
(c) arithmetic mean
(d) none of the above

(iii) The algebraic sum of deviation of a set of n values from A.M. is
(a) n
(b) 0
(c) 1
(d) none of the above

[Ans. (i) b (ii) c (iii) b]

3. Comment whether the following statements are true or false.
(i) The sum of deviation of items from median is zero.
(ii) An average alone is not enough to compare series.
(iii) Arithmetic mean is a positional value.
(iv) Upper quartile is the lowest value of top 25% of items.
(v) Median is unduly affected by extreme observations.
[Ans. (i) False (ii) True (iii) False (iv) True (v) False]

4. 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 Rs)                       0-10 10-20 20-30 30-40 40-50 50-60
Number of retail shops                                  12     18      27        -         17      6

(Ans. The value of missing frequency is 20 and value of the median is Rs 27.41) 5. 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 Rs) 120 150 180 200 250 300 220 350 370 260

(Ans. Rs 240)