1. Introduction Central Tendency 1 Central Tendency or more simply average is a measure of finding a representative single figure for a large set of data for making meaningful comparisons with others sets or body of data. Such an average is somewhere within range of data and is therefore a measure of central tendency. Some of the techniques that can be used for measuring Central Tendency are given below- 1.Arithmatic Mean 2. Median 3. Mode 4. Geometric Mean 5. Harmonic Mean 6. Weighted Mean These tools can be used for statistical population or samples depending upon type of data. “An average is a single figure that represents the whole group.”-Clark “An average is a typical value that represents all the individual values in a series.”-Croxton and Cowden Meaning Definitions

2. Purpose & functions Central Tendency 2 1.Brief and Simple- It can present raw data which may be unorganised and complex into a simple and systematic form reducing the mass of data into single figure to draw conclusions. 2.Helpful in comparison- It helps in making comparison of different sets of data. For e.g average age of male population in India may be compared with similar statistic in UK to understand demographic patterns. 3.Helpful in policy formulation – Once inferences are drawn by use of average, social, economic, educational, industrial and other policies can be formed to take corrective actions to improve conditions. 4.Basis of statistical analysis- Average figure can put focus on general patterns, interests, orientation of a large population and give an inspiration for further study of drivers of such interest. 5. Representation of Universe- It can provide a single representative figure for a large data set or for sub components for understanding general position or state of affairs.

3. Characteristics & types Central Tendency or Average 3 Good Average Types of average Characteristics of good average are- 1.It should be simple to compute. 2.Should be easy to understand. 3.Should be uniquely defined. 4.Should be based on all observations without unduly affected by extreme observations. 5.Should be capable for further algebraic treatment. Types of Averages Mathematical Averages Arithmatic Mean Geometric Mean Harmonic Mean Positional AveragesMedian Mode

4. Arithmatic Mean Central Tendency 4 Individual series Arithmatic mean is most popular and widely known measure of central tendency. It is defined as value arrived at by adding all items of a series and then diving it by total number items. It may be divided into 2 types- 1.Simple arithmatic mean 2.Weighted arithmatic mean In case of individual series, arithmatic mean can be computed by applying any of 2 methods a.Direct method Formula Where is arithmatic mean ∑x is values of items of a series n is number of observations contd…. Definition

5. Arithmatic Mean Central Tendency 5 Discrete series Short Cut method When number of observations are large, arithmatic mean can be calculated by short cut method. The deviations are taken from an assumed mean. The formula to calculate it is- Where ‘d’ is deviations of items from assumed mean and ‘A’ is assumed mean. To calculate Arithmatic mean in discrete series, 3 methods may be used 1.Direct method 2. Shortcut method 3. Step deviation method --------------------------------------------------------------------------------- The following formula is used for direct method- Where f is frequency, x is value of variables and N is total number of observations. Formula for Shortcut method Where A is assumed mean, Efdx=Ef(X-A) and N is sum of frequency contd….

6. Central Tendency 6 Step Deviation method Formula used in this method is Where C is the common factor. For calculating arithmatic mean for continuous series, 3 methods are used 1.Direct 2. short cut 3. Step deviation Direct- Short Cut- Step Deviation- Continuous series

7. Arithmatic Mean (AM) Central Tendency 7 1.The sum of deviations of items from AM is always zero. 2.The sum of squared deviations of the items from AM is minimum i.e. it is less than squared deviation of items from any other value. 3.If each item of series is 4.The product of Merits- 1.It is easy to calculate and simple to understand. 2.It is based on all observations. 3.It is rigidly defined and based on calculated value rather than the positional value. Demerits- 1.Its outcome is affected by extreme values. 2.It cannot be used inn an open series. 3.Sometimes gives confusing results like number of children born to population may be presented in decimal points. Properties Merits & demerits

8. Median Central Tendency 8 Median is another important measure of central tendency based on the positional average. It is defined as middle value of series when the series is arranged in either ascending or descending order. “The median is that value of the variable which divides the group into two equal parts, one part comprising all value greater and the other values less than the median.”-Conner Calculation of Median M=size of {N+1}th item 2 Odd number series- If the number of items are odd, then the medium is middle value after the items are arranged in ascending or descending order. Even number items- In case of even number of items, medium is arithmetic median of 2 middle items after the items are arranged in ascending or descending order of their magnitude. Introduction Definition Individual Series

9. Median Central Tendency 9 Calculation of Median M=size of {N+1}th item 2 1.It is easy to understand and compute. 2.Median is not affected by extreme values. 3.Median is most suitable for finding qualitative aspects and also provides most appropriate average in open ended classes. 4.It is rigidly defined. Discrete Series Continuous Series Mid value Series Merits

10. Median & Partition values Central Tendency 10 1.It requires arranging of data but other averages do not need this. 2.Not based on all the observations of the series so it is a positional average. 3.It cannot be computed correctly if the numbers of items in the series is even. 4.It is difficult to calculate if the items in a series are very small or very large. Just as Median divides the series into 2 equal parts, there are other measures which divides the series into 4, 10 and 100 equal parts. These are called quartiles, deciles and percentiles. Quartiles-Divides the series into 4 equal parts and are denoted by Q1, Q2 and Q3 for any series. Deciles- Divides the series into 10 equal parts and every series have nine deciles denoted by D1 through D9. Percentiles- Divides the series into 100 equal parts and every series have ninety nine deciles denoted by P1 through P99. Demerits Partition Values

11. Mode Central Tendency 11 Mode is another important measure of central tendency and is defines as value which appears most frequently in the series. “The value of the variable which occurs most frequently in distribution is called the mode.”- Kenny and Keeping “Mode is the value which has greatest frequency density.”- A.M.Tuttle. In case of individual series mode can be computed by 2 methods 1. Inspection method-This involves inspecting the series to find most frequently occurring value which is mode. 2. By changing individual series into discrete series-When the number of items is very large, individual series is first converted into discrete series to find the value corresponding to which there is highest frequency. Introduction Definition Individual Series

12. Mode Central Tendency 12 Under this Mode can be calculated using 2 methods- 1.Inspection method- Mode is determined by inspection only. By the method is generally used where frequency increase upto a point and decrease after reaching a maximum point. 2. Grouping method- Discrete Series

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