0
Chapter 2: Measure Of Central Tendency:
- The numerical value of an observation (also called central value) around which most numerical values of other observations in the data set show a tendency to cluster or group, called the central tendency.
- The extent to which numerical values are dispersed around the central value, called variation.
- The extent of departure of numerical values from symmetrical (normal) distribution around the central value, called skewness.
These three properties – central tendency, variation and shape of the frequency distribution – may be used to extract and summarize major features of the data set by the application of certain statistical methods called descriptive measure of summary measures.
There are three types of summary measures:
- Measure of central tendency. (discussed in this chapter)
- Measure of dispersion or variation. (chapter - 3)
- Measure of symmetry – skewness. (chapter - 4)
Measure of Central Value
One of the most important objective of statistical analysis is to get one single value that describes the characteristic of the entire mass of unwieldy data. Such a value is called the central value or an ‘Average’ or the expected value of the variable.
“Average is an attempt to find one single figure to describe whole of figures” – CLARK
“The average is sometimes described as a number Which is typical of the whole group” – Leabo
Objective of Averageing:
- To get single value that describes the characteristic of the entire group.
- To facilitate Comparison Measures of central value, by recucing the mass of data to one single figure, enable comparison to be made.
Features of a good Average:
- It should be rigidly defined so that different persons may not interpret it differently.
- It should be easy to understand and easy to calculate.
- It should be based on all the observations of the data.
- It should be easily subjected to further mathematical calculations.
- It should be least affected by the fluctuation of the sampling.
- It should not be unduly affected by the extreme values.
- It should be easy to interpret.
- It should have sampling stability. It means that if the average is computed for similar groups, the result should also be similar.
The various measures of central tendency or average can be classified in the following categories:
| Mathematical Average. These averages are mathematical in nature and deal with those characteristics of a data set which can be directly measured quantitatively, such as: Income, Profit, Level of production, Rate of groth, etc. |
| Averages of position These averages are refer to position of the value of an observations of interest rather than computing it and it measure qualitative characteristics of a data set, such as: Honesty, intelligence, Beauty, Consumer acceptance, and so on |
|
(a) Arithmetic Mean (A.M.) -Simple -Weighted (b) Geometric Mean (G.M.) (c) Harmonic Mean (H.M.) |
|
(a) Median (b) Quartiles (c) Deciles Partition Values (d) Percentiles (e) Mode |
Note: An average (Mean, G.M., H.M., Median, Mode) is the representative of whole series, but quartile, deciles, percentiles are averages of parts of the distribution (Series).
Mathematical Averages:
(a) Arithmetic Mean:
|
| Direct Method | Short-cut Method | Step-deviation Method |
| Individual Observation | Where n = No. of observations.
| ----- | ----- |
| Discrete Series | Where
| Where , A = assumed mean. d = X - A *
| ----- |
| Continuous Series: | Where , X = mid value of class
| Where , A = assumed mean. d = X - A, X = mid value of class
| Where , A = assumed mean, d = X – A, X = mid value of class, C = Class length.
|
Note: 1. Any value whether existing in data or not can be taken as the assumed mean and the final answer would be the same. However the nearer the assumed mean is to the actual mean, the lesser are the calculations.
- If data is continuous then to find mean there is no need to convert inclusive data into exclusive.
- When data are given by the inclusive method it is not necessary to adjust the class for calculating arithmetic mean because the mid-points remain the same whether or not the adjustment is made. However, in case of median and mode adjustment is necessary.
Properties:
- The sum of the deviations of the items from the arithmetic mean is always zero.
That is :
- The sum of the squared deviations of the items from arithmetic mean is minimum.
That is is minimum if A = .
- If the set of n observations x1, x2, . . ., xn have its mean then
- For some constant b, mean of bx1, bx2, . . ., bxn is .
- For some constant p, mean of ,. . ., is .
- For some constant a, mean of x1± a, x2± a, . . ., xn± a is
Weighted Arithmetic Mean:
One of the limitations of the arithmetic mean discussed above is that it gives equal importance to all the items. But there are cases where the relative importance of the different items is not the same. When this is so, we compute weighted arithmetic mean. The term ‘weight’ stands for the relative importance of the different items. The formula for computing weighted arithmetic mean is: and for frequency distribution:
| Mean. | Uses: Estimates are always obtained by mean |
| Merits | 1. Neither the arraying of data as required for calculating median nor grouping of data as required for calculating mode is needed while calculating mean. 2. It is based on all values given in the data set. 3. It is calculated value, and not based on position in the series. 4. Every data set has one and only one mean. |
| Demerits | 1. Since the value of mean depends upon each and every item of the series extreme items, i.e. very small and very large items, undualy affect the value of the average. 2. In a distribution with open-end classes the value of mean can not be computed without making assumptions regarding the size of the class interval of the open-end classes. 3. A.M. Can not be calculated accurately for unequal and open-ended class Interval. 4. In extremely asymmetric (Skewed) distribution, usually AM is not representative of the distribution and hence it is not a suitable measure of location |
Examples:
- A person walks 9 hours at a speed of 3 kms per hour and again walks 8 hours at a speed of 4 kms per hour. Find the average speed per hour by using appropriate average.
- The mean age of a combined group of men and women is 30 years. If the mean age of the group of men is 32 and that of the group of women is 27, find out the percentage of men and women in the group. (Hint: combined mean. Answer: N1 = 60% and N2 = 40%)
- A scooterist purchased petrol at the rate of Rs. 24, Rs. 29.5 and Rs. 36.85 per litre during three successive years. Calculate the average price of petrol if he purchased 150, 180 and 195 liters of petrol in the respective years. (Hint: Weighted mean. Answer: Rs. 30.66)
(b) Geometric Mean:
| Individual Observation | If the set of n observations x1, x2, . . ., xn are given then OR
|
| Discrete Series | If the set of n observations x1, x2, . . ., xn with frequencies f1, f2, . . ., fn are given then OR
|
| Continuous Series: | If the set of middle values of n intervals are x1, x2, . . ., xn with frequencies f1, f2, . . ., fn are given then OR |
| G.M. | Uses: 1. To find the rate of population growth and the rate of interest. 2. In the Construction of Index numbers. |
| Merits | 1. It is based on all the observations. 2. It is not affected much by fluctuations of sampling. 3. It gives comparatively more weight to small items. |
| Demerits | 1. If any one of the observations is zero geometric mean becomes zero. 2. If any one of the observations is negative it will be imaginary. 3. It can not be obtained by inspection |
Examples:
- The rate of increase in population of a country during the last three decades is 5%, 8% and 12%. Find the average rate of growth during the last three decades. (Hint: Use G.M. and answer is 8.2, BJKS-102
- A machinery is assumed to depreciate 44% in value in the first year, 15% in the second year, and 10 % per year for the next three years, each percentage being calculated on diminishing value. What is the average percentage of depreciation for the entire period? (Hint: Apply G.M., BJKS-105)
- An economy grows at the rate of 2% in the first year, 2.5% in the second year, 3% in the third year, 4% in the fourth year, . . . and 10% in the tenth year. What is the average rate of growth of the company? (Hint: Apply G.M.: answer is 5.6%, BJKS-128)
- Compared to the previous year, the overhead expenses went up 32% in 1994, increased by 40% in the next year, and by 50% in the following year. Calculate the average rate of increase in overhead expenses over the three years.
(Hint: Apply the formula Pn = P0 (1 + r)n: answer is 40.5%, BJKS-105)
(c) Harmonic Mean:
| Individual Observation | If the set of n observations x1, x2, . . ., xn are given then
|
| Discrete Series | If the set of n observations x1, x2, . . ., xn with frequencies f1, f2, . . ., fn are given then
|
| Continuous Series: | If the set of middle values of n intervals are x1, x2, . . ., xn with frequencies f1, f2, . . ., fn are given then
|
| H.M. | Uses: 1. It is useful average when we deal with average of rates. 2. It is useful averaging time rates. 3. It is used when rates are expressed as x per y, where x is constant. |
| Merits | 1. It is based upon all the observations. 2. While calculating H.M., more weightage is given to smaller values in a data set because in this case, the reciprocal of given value is taken for the calculation of H.M. |
| Demerits | 1. It can not be obtained by inspection 2. It is not often used for analyzing business problems. 3. If any one of the observations is negative and / or zero it can not be calculated. |
Note: Geometric mean and harmonic mean are known as ratio averages as they are most appropriate where the data comprise rates, ratios or percentages instead of actual quantities. Geometric mean is to be used while dealing with rates and ratios. Harmonic mean mean is to be used in compiling special types of average rates or ratios, where time factor is variable and the act being performed, e.g., distance is constant.
Examples:
- A scooterist purchased petrol at the rate of Rs. 24, Rs. 29.5 and Rs. 36.85 per litre during three successive years. Calculate the average price of petrol if he spends Rs. 3850, Rs. 4675 and Rs. 5825 in the three years. (Answer: Rs. 30.09) <
