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MEASURES OF CENTRAL TENDENCY AND MEASURES OF DISPERSION

Central tendency refers to typical or average values in a data set or probability distribution. The three most common measures of central tendency are the mean, median, and mode. The mean is the average calculated by summing all values and dividing by the total number. The median is the middle value when data is arranged in order. The mode is the most frequently occurring value. Other measures discussed include range, which is the difference between highest and lowest values, and quartiles, which divide a data set into four equal parts based on the distribution of values.

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Central tendency is a central value or a typical value for a probability distribution. It is occasionally called an average or just the center of the distribution. The most common measures of central tendency are the arithmetic mean, the median and the mode. A central tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Occasionally authors use central tendency (or centrality), to mean "the tendency of quantitative data to cluster around some central value."
• ARITHMETIC MEAN • HARMONIC MEAN • GEOMETRIC MEAN • MEDIAN • MODE
 IT SHOULD BE RIGIDLY DEFINED. IT SHOULD BE EASY TO UNDERSTAND AND CALCULATE  IT SHOULD BE BASED ON ALL THE OBSERVATIONS OF DATA .  IT SHOULD BE SUBJECTED TO FURTHER MATHEMATICAL CALCULATIONS  IT SHOULD BE LEAST AFFECTED BY FLUCTUATION OF SAMPLING
ARITHMETIC MEAN HARMONIC MEAN GEOMETRIC MEAN
The arithmetic mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers.
INDIVISUAL SERIES DISCRETE SERIES
IT ISEASY TO UNDERSTAND AND CALCULATE.  BASED ON ALL THE ITEMS OF SAMPLE  RIGIDLY DEFINED BY MATHEMATICAL FORMULA.  WE CAN COMPUTE COMBINED ARITHMETIC MEAN.  IT HAS SAMPLING STABILITY .  AFFECTED BY EXTREME VALUES  NOT USEFUL FOR STUDYING QUALITATIVE PHENOMENON
The median is also a frequently used measure of central tendency. The median is the midpoint of a distribution: the same number of scores is above the median as below it.
INDIVISUAL SERIES WHEN N IS EVEN ARRANGE DATA IN INCREASING ORDER OR DECREASING ORDER TAKE THE ARITHMETIC MEAN OF MIDDLE VALUES ie N/2 +N+1/2 WHEN N IS ODD ARRANGE DATA IN INCREASING OR DECREASING ORDER
(1) It is very simple to understand and easy to calculate. In some cases it is obtained simply by inspection. (2) Median lies at the middle part of the series and hence it is not affected by the extreme values. (3) It is a special average used in qualitative phenomena like intelligence or beauty which are not quantified but ranks are given. Thus we can locate the person whose intelligence or beauty is the average. (4) In grouped frequency distribution it can be graphically located by drawing ogives.
(1)In simple series, the item values have to be arranged. If the series contains large number of items, then the process becomes tedious. (2) It is a less representative average because it does not depend on all the items in the series. (3) It is not capable of further algebraic treatment. For example, we can not find a combined median of two or more groups if the median of different groups are given. (4) It is affected more by sampling fluctuations than the mean as it is concerned with on1y one item i.e. the middle item
A statistical term that refers to the most frequently occurring number found in a set of numbers. The mode is found by collecting and organizing the data in order to count the frequency of each result. The result with the highest occurrences is the mode of the set.
•It is easy to understand and simple to calculate. •It is not affected by extreme large or small values. •It can be located only by inspection in ungrouped data and discrete frequency distribution. •It can be useful for qualitative data. •It can be computed in open-end frequency table. •It can be located graphically. •It is not well defined. •It is not based on all the values. •It is stable for large values and it will not be well defined if the data consists of small number of values. •It is not capable of further mathematical treatment. •Sometimes, the data having one or more than one mode and sometimes the data having no mode at all.
A statistical term describing a division of observations into four defined intervals based upon the values of the data and how they compare to the entire set of observations. Each quartile contains 25% of the total observations. Generally, the data is ordered from smallest to largest with those observations falling below 25% of all the data analyzed allocated within the 1st quartile, observations falling between 25.1% and 50% and allocated in the 2nd quartile, then the observations falling between 51% and 75% allocated in the 3rd quartile, and finally the remaining observations allocated in the 4th quartile.
Question 1: Find the quartiles of the following data: 3, 5, 6, 7, 9, 22, 33. Solution: Here the numbers are arranged in the increasing order, n = 7 Lower quartile, Q1 = n+1th4 item = (7+1)4 item = 2nd item = 5 Median, Q2 = n+1th2 item = (7+1)2 item = 4th item = 7 Upper Quartile, Q3 = 3 n+1th4 item = 3(7+1)4 item = 6th item = 22
Question 2: Find the Quartiles of the following marks:21, 12, 36, 15, 25, 34, 25, 34 Solution: First we have to arrange the numbers in the ascending order. 12, 15, 21, 25, 25, 34, 34, 36 n=8 Lower Quartile, Q1 = n+1th4 item = 8+14 item = 2.25th item = 2nd item + 0.25(3rd item - 2nd item) =15 + 0.25(21 - 15) = 15 + 0.25(6) = 16.5 Second Quartile, Q2 = n+1th2 item = 8+12 item = 4.5th item = 4th item + 0.5(5th item - 4th item) =15 + 0.5(25 - 15) = 15 + 0.5(10) = 20 Third Quartile, Q3 = 3n+1th4 item = 3(8+1)4 item
The range is very easy to calculate because it is simply the difference between the largest and the smallest observed values in a data set. Thus, range, including any outliers, is the actual spread of data.
Range = difference between highest and lowest observed values
Example: In {4, 6, 9, 3, 7} the lowest value is 3, and the highest is 9. So the range is 9-3 = 6.
The range is an informative tool used as a supplement to other measures such as the standard deviation or semi-interquartile range The range value of a data set is greatly influenced by the presence of just one unusually large or small value (outlier). The disadvantage of using range is that it does not measure the spread of the majority of values in a data set—it only measures the spread between highest and lowest values. As a result, other measures are required in order to give a better picture of the data spread.
MEASURES OF CENTRAL TENDENCY AND MEASURES OF DISPERSION