Descriptive statistics

Measures of Central Tendency

The tendency of the variate values to get clustered around a certain representative value of the series is known as central tendency.

The value of the variate which is thoroughly representative of the series is referred to as the central tendency .

Various methods of determining the actual value at which the variate values of the data tend to cluster are called the measures of central tendency.

Arithmetic mean or average

Arithmetic mean may be defined as the quotient obtained by dividing the total of all the values of a variable with the total number of observations.

If X₁, X₂, X₃, ..........., X_n are the individual observations pertaining to variable X, then the arithmetic mean is obtained as:

Σ X_i

`X = -------

n

Where,

`X = Arithmetic mean

X_i = i^th value of variable X

Σ = sum of values of X from 1^st to n^th

n = Total number of observations or items.

Note: The unit of measurement should be mentioned with the value of the mean.

Methods of Computation

1. Direct Method
2. Short-cut Method
3. Step Deviation Method, and
4. Shortest Method

Direct method

Where,

ΣfX = Sum of the product of the values and their corresponding frequencies.

Σfm = Sum of the product of mid-values of classes and their frequencies.

n = Σf = Total number of observations.

Short-cut method (deviation method or assumed mean method )

1. Choose an "arbitrary" or "assumed" mean, say A.
2. Subtract A from each of the variate values (Xi) to get their deviations, i.e., d = (X – A).
3. Obtain algebraic sum of deviations giving due consideration to their signs (+ or - ).
4. Add the mean of these deviations to the assumed mean to obtain the arithmetic mean.

Where,

A = Assumed mean

d = (X – A) or (m – A) = Deviation of individual items or mid-values from assumed mean

Σd = Sum of deviations of individual observations (or mid values of the classes) from the

assumed mean

Σfd = Sum of the products of the deviations and their corresponding frequencies

Step deviation method

Where,

C = Common factor (or constant) by which each deviation is divided, and

Shortest method

Where,

X = Last value of the variable arranged in ascending order for individual or discrete series.

m = Mid-value of the last class arranged in ascending order for continuous series.

i = Interval of equal magnitude between X or m-values depending upon the series used.

Geometric Mean

When data on increase in the ratio or percentage, the geometric mean is suitable Measure of Central Tendency.

Suppose we have an investment which returns 10% the first year, 50% the second year, and 30% the third year. What is its average rate of return? It is not the arithmetic mean, because what these numbers mean is that on the first year our investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.50, and during the third year it was multiplied by 1.30.

Geometric mean can be defined as the n^th root of the product of values of the given set of data.

Methods of Computation

1. Root Method

The given formula is appropriate when the values of the items are small, simple and capable of being factorized easily. If the values are big or complicated, then the geometric mean is more easily calculated by the logarithmic operation method

2. Log method

Direct Method

Short cut method

Where,

Log A = Log of the assumed mean,

Σd (log A) = Sum of deviation of log X and log A = (log X – log A)

Properties of Geometric Mean

1. For any series, geometric mean is smaller than arithmetic mean. However, when all the values in a series are equal then geometric mean = arithmetic mean.
2. If any value of a series is zero, the value of geometric mean is infinity (not defined).
3. Geometric mean can not be calculated if the total number of negative values is odd.
4. The geometric mean raised to the nth power equals to the product of values:

GMn=X1.X2......Xn

5. The geometric mean of different series with same n and same product is always equal,

e.g. geometric mean of (1, 36), (2,18) and (4, 9) = √36 = 6.

6. The product of a series of numbers remains the same even when each number of the series is replaced by its geometric mean.

Use of Geometric Mean

• For averaging the ratios and percentages.
• To find the average of index numbers.
• When it becomes necessary to give more weights to small items and less weights to big items.
• In the gradual increase or decrease of a given value or a number over certain period of time, e.g. in finding compound interest or depreciation costs.

Merits of Geometric Mean

• It is rigidly defined.
• It is based on all the observations.
• It gives better results when we have to calculate percentage increase in any variable.

Demerits of Geometric Mean

• It is neither easy to compute nor easy to understand.
• If any observation is zero then geometric mean is zero.
• It can not be calculated if any observation is negative.

Harmonic mean

The Harmonic Mean is defined as the reciprocal of the arithmetic mean of the reciprocals of the values of a variable.

Methods of Computation

Direct Method

Short cut method

Where, Rec. A = Reciprocal of the assumed mean.

Uses of Harmonic Mean

1. When frequency distribution is skewed to the right.
2. In case of graphic representation of data.
3. In computation of average rate of speed, average rate of price, average rate of income or decrease in profit etc.

Merits of Harmonic Mean

• It is rigidly defined.
• It is based on all observations.
• It is capable of further algebraic treatment.
• It can be calculated even when the series has any negative value.
• It converts a skewed distribution to normal.
• It gives a straighter curve than that of arithmetic or geometric mean.

Demerits of Harmonic Mean

• It is not easy to understand by an ordinary man.
• Its calculation is cumbersome.
• It can not be calculated if any of the item is zero.

Relation between Arithmetic, Geometric and Harmonic means

For any series AM ≥ GM ≥ HM. But if all the observations are same, then these three means are equal.

For two observations (X and Y):

Weighted Mean

If X1, X2, .........., Xn denote n values of a variable X and W1, W2, ......., Wn are their weights respectively, then their weighted mean is given by,

Uses of Weighted Mean

1. Weighted mean is used when the number of individuals in different classes of a group is varying widely.
2. When the importance of all the items in a series is not same.
3. When the ratios, percentages or rates ( e.g., quintals per acre, rupees per kg or rupees per litre etc.) are to be averaged.
4. When the average of a series is to be obtained from the averages of its components.
5. It is particularly used in the calculation of birth rates, death rates, index numbers, average yield of crops, etc.

MEDIAN

The median can be defined as the value of the middle item of a series arranged in ascending or descending order of magnitude.

Computation Methods

1. Tabular Method

The value of the median is located from table as follows:

• Individual Series

(i)When n = Odd:-

Median = (n+1)/2 ^th item in an array

When n = even :-

Median = Average of (n/2)^th and (n+2)/2^th item

• Discrete Series

1. Calculate cumulative frequencies
2. Obtain the value of ( (n + 1)/2)th item.
3. Take the value corresponding to this item in relation to the cumulative frequency as the median.

• Continuous Series

When all the classes are of equal length

Where, L0 = Lower limit of median class, L1 = Upper limit of median class

f = Frequency of median class, Cf = Cumulative frequency up to the class preceding the median class, and i = Class interval

When the classes are not of equal length

2. Graphic method

Method-I

1. Arrange the value column in “less than” order along with their corresponding cumulative frequencies in an ascending order.
2. Draw the first quadrant of the graph taking both the horizontal and vertical axes in the positive direction.
3. Arrange the frequencies along the vertical axis on a natural scale so as to accommodate the maximum cumulative frequency on it.
4. Plot the different points of intersection of the values and their respective frequencies on the graph paper and then draw the curve in free hand manner. Such a curve is called “less than ogive”.
5. Compute the median as (n/2).
6. With reference to the required item, i.e. median, draw a perpendicular from the vertical axis on the ogive curve and then from the point of touching the ogive curve draw another perpendicular to the horizontal axis. The point thus located on the horizontal axis will indicate the median.

Method-2

We can obtain the median by drawing both less than and more than ogive in the same curve simultaneously. The point of intersection of both the cumulative frequency curves with respect to variate value on the horizontal axis will be the median of given continuous frequency distribution.

Example: Determine the median from the following frequency distribution by graphic method and compare the same with that obtained from the formula.

Method I

Calculation from the formula

Median number = (n/2) = (56/2) = 28

(n/2) - cf

Median = L₀ + ------------- X i

f

28 – 20

Median =40 + ----------- X 10 = 40 + 5.33 = 45.33

15

Properties of Median

1. The sum of absolute deviations of individual values from the median is minimum as compared to deviations taken from any other measures of central tendency.
2. The median divides the series into two equal halves. Thus it has the property that half of the values in the series exceed it and half fall short of it.

MODE

The mode is defined as the value of a series that occurs maximum number of times, i.e. the item or value having maximum frequency.

Methods of Determination

1. Method of inspection

Mode or the modal class is determined by simple inspection of the distribution.

The value occurring maximum number of times or the class value against which the maximum frequency stands is taken as the modal value or the modal class.

2. Method of Grouping

(a) Preparation of grouping table

1. Put the values (or classes) in the left hand column and their corresponding frequencies in another adjacent column to its right. Mark the frequency column as column number 1.
2. In column 2, group the frequencies in twos beginning with the first frequency.
3. In column 3, group the frequencies again in twos but beginning with the second frequency.
4. In column 4, group the frequencies in threes beginning with the first frequency
5. 5.In column 5 group in threes beginning with second, and
6. Lastly in the 6^th column group in threes beginning with the third frequency.
7. After groupings are over, identify the maximum figures of each of the 6 columns.

(b) Preparation of an analysis table

1. Probable modal values are stated in the top horizontal line under the different columns and the different column numbers of grouping table are put at the left hand side of the table.
2. The values showing the maximum grouped frequencies in the grouping table are identified by mark in their respective columns.
3. At the bottom of the table, the number of such marks put under the probable value column is totaled.
4. The probable value showing the maximum of such total is then considered as the modal value or the modal class as the case may be.

Interpolation of Mode from a Continuous Series

After determining the modal class, the exact value of the mode is obtained by applying any one of the following formula of interpolation:

3. Graphic Method

Suitable in case of frequency distribution of a continuous nature.

Procedure:

1. Draw a histogram with the given class intervals and their frequencies.
2. The highest rectangular (or bar) is identified from the histogram. This rectangular is the modal rectangular.
3. Two lines are drawn to this modal rectangular diagonally starting from each of its upper corners and ending with the nearest corner of their respective adjacent rectangular.
4. Then from the intersecting point of these two diagonal lines a perpendicular is drawn to the base, i.e., X-axis. The point at which the perpendicular touches the base is identified as the modal value.

4. Method of Empirical Relation

This method, proposed by Karl Pearson, is based on the pertinent relationship among the mean, median and mode in case of moderately asymmetric or skewed distribution.

Mode = 3 Median - 2 Mean

Merits of Mode

1. It is easy to understand and easy to calculate.
2. Neither the extreme values are needed in its computation nor they affect it.
3. It can be obtained simply by inspection also.
4. It can also be obtained graphically.
5. It gives the most representative value of the series.
6. It can be determined straight way from an open-end series.
7. It is capable of studying qualitative data also as its determination depends up on the frequencies rather than the value of the item.
8. It is a reliable measure for studying skewness of a distribution.

Demerits of Mode

1. It is not based on all the observations.
2. It is not capable of further algebraic treatment.
3. In continuous and bimodal series, its determination involves many steps.
4. It can not be determined in a series with unequal class intervals unless they are made of uniform length.
5. It is not useful in the cases where it is desirable to give weights to extreme values.
6. It is the most ill defined average.

Uses of Mode

1. It is very much useful in the field of business and commerce. It helps the businessmen in taking a decision to procure or produce items in large quantities to increase his sale and profit.
2. Meteorological and also business forecasting are based on the modal value.

Relationship between Mean, Median and Mode

1. In a symmetrical distribution measures Mean = Median = Mode
2. In a positively (right skewed) distribution Mean > Median > Mode
3. In the negatively (left skewed) distribution Mean < Median < Mode