Measures of Central Tendency

Central Tendency “A measure of central tendency or an average is a figure (value) that represents the entire group of data.” “An average value is a single value within the range of the data that is used to represent all the values in the series.” Since an average is somewhere within the range of the data, it is called a measure of central tendency.

Central Tendency Objectives and Functions: To present diversified and complex data in a summarized form. To facilitate comparison. To facilitate decision making. To establish relationship. To facilitate statistical analysis.

Central Tendency Essentials of a Good Average: Clearly defined. Representative. Easy to understand. Capable of further algebraic treatment. Not affected by extreme values. Absolute number.

Mathematical averages Central Tendency Averages Mathematical averages Arithmetic Mean Geometric Mean Harmonic mean Positional averages Median Partition Values Mode

Arithmetic Mean Arithmetic mean is the most commonly used measure of central tendency. Definition: “Sum of the values of all observations divided by the number of observations.” Arithmetic Mean Simple Arithmetic Mean Weighted Arithmetic mean

Arithmetic Mean In Simple Arithmetic Mean, all the observations in a series are given equal weightages. In weighted Arithmetic Mean, weights are assigned to various items according to their importance.

Simple Arithmetic Mean Calculation: There are 3 methods for all the 3 series

Weighted Arithmetic Mean Application: When all the items in the series are not of equal importance. When assigning weights to different items according to their relative importance is necessary. For Example, If the impact of price rise is to be observed for any commodity. Calculation:

Arithmetic Mean Important Properties: The sum of deviations of the observations from their arithmetic mean is always zero. Sum of the square of the deviations of the observations from their Arithmetic mean is minimum. If the arithmetic mean of two or more series is known, combined mean of all the series can be calculated

Arithmetic Mean If each observation in a series is increased or decreased by a constant, then the arithmetic mean of the new series will also be increased or decreased by the same constant. Arithmetic mean of following 5 observations: 5, 7, 3, 4, 11 If we add 4 in each observation, then we get the new observations as: 9, 11, 7, 8, 15 Thus, the new mean is 10 i.e. the old mean gets increased by 4.

Arithmetic Mean If each observation in a series is multiplied or divided by a constant, then the arithmetic mean of the new series will also be multiplied or divided by the same constant. Arithmetic mean of following 5 observations: 5, 7, 3, 4, 11 If we multiply 2 in each observation, then we get the new observations as: 10, 14, 6, 8, 22 Thus, the new mean is 12 i.e. the old mean gets multiplied by 2.

Arithmetic Mean MERITS: It is simple. It is a representative value. It is rigidly defined. It is capable of algebraic treatment. It is a stable measure It is least affected by fluctuations of sample.

Arithmetic Mean DEMERITS: It is affected by extreme values. Mathematically calculated More importance to higher values Mean value may not exist in the series.

Positional Averages “Positional Averages determine the position of variables in a given sets of data.” Positional Averages Median Partition Values Mode

Median “Median is that point on the scale of score below which one half of the score lies and above which one half of the score lies.” “The median is that value of the variable which divides the group into two equal parts, one part comprising all values greater than the median value and the other part comprising all the values smaller than the median value.“

Median Calculation:

Median Graphically:

Median MERITS: Simplicity Certainty Unaffected by extreme values Graphic Presentation Possible in case of incomplete data Practical application

Median DEMERITS: Further algebraic treatment not possible Lack of representative Nature Requires a lot of efforts Unrealistic assumptions Affected by variations in the sampling

Partition Values Median divides in equal 2 parts. If more than 2 parts are required for any distribution, following Partition values are used. Quartile – 4 parts Deciles – 10 parts Percentiles – 100 Parts

Partition Values Quartile: Deciles: Percentiles: Divides into 4 equal parts. Three quartiles are there, Q1 – 1st or Lower Quartile (25%) Q2 – 2nd or Middle Quartile (50%) = Median Value Q3 – 3rd or Upper Quartile (75%) Deciles: Divides into 10 equal parts. Nine deciles are there, D1 to D9. Also, D0 = Minimum value D5 = Median = Q2 D10 = Maximum Value Percentiles: Divides into 100 equal parts. Ninety nine Percentiles are there, P1 to P99. Also, P0 = Minimum value P50 = Median = Q2 = D5 P100 = Maximum Value

Partition Values Calculation:

Mode “The mode of a distribution is the value at the point around which the items tend to be the most heavily concentrated.” It is the most frequently observed data value. It is the position of greatest density or highest concentration of value. It is the most common value found in the series. It is denoted by Z.

Mode Calculation: For Continuous Series two Methods are used, Inspection or Observation Method Grouping Method Also Mode can be found graphically for continuous series using Histogram.

Mode Graphically:

Mode MERITS: Simple Unaffected by very small values Graphic Presentation Representative in Nature Useful in open ended class intervals Exact value

Mode DEMERITS: Not rigidly defined Difficult to identify Not based on all the observations Not capable of algebraic treatment Difficult to calculate Not suitable for assigning weights

Relationship among mean, median and mode Symmetrical distribution: The observations are equally distributed. The values of mean, median and mode are always equal. i.e. mean = median = mode Asymmetrical distribution: The observations are not equally distributed. two possibilities are there: Positively Skewed  Mean > Median > Mode Most values fall to the right of the mode i.e. mode is minimum Negatively Skewed  Mean < Median < Mode Most values fall to the left of the mode i.e. mode is maximum In general, the relation can be given by, Mode = 3 median – 2 mean

Relationship among mean, median and mode

Suitability Mean: Median: Mode: It is commonly used as it is simple, easy to calculate and precisely defined. It takes all the items into consideration. Also cancels all the negative deviations with positive deviations. Median: Used to measure qualitative aspects like intelligence, honesty, beauty etc. Mode: Most commonly used in business sphere. It helps in measuring the popularity of goods, ascertain rainfall or temperature in any region.