Measures of Central Tendency

Measures of Central Tendency K.THIYAGU, Assistant Professor DoE, Central University of Kerala, Kasaragod @ThiyaguSuriya 1 Measures of Central Tendency The central tendency is stated as the statistical measure that represents the single value of the entire distribution or a dataset. A measure of central tendency is a single value that attempts to describe a data set by identifying the central position within that set of data. @ThiyaguSuriya 2 Characteristics Measures of central tendency are sometimes called measures of central location. @ThiyaguSuriya A single number that represents the entire set of data (average) 3 Central Tendency Mean Median Mode Average Value Middle Value Most Common Value @ThiyaguSuriya 4 @ThiyaguSuriya 5 @ThiyaguSuriya 6 Requisites of Measures of Central Tendency 3. Least affected by Fluctuations of Sampling 4. Not Affected much by Extreme Values The value of an average should not be affected by just one or two very large or very small items, There should be sampling stability in an average. 2. Easy to Understand and Calculate 5. Based on all the Observations An average should be based on all the observations, The value of an average should be computed using a method that is simple 1. Rigidly Defined 6. Capable of further Algebraic Treatment An average should be rigid and clear. A good average should have the capability of further statistical and mathematical calculations @ThiyaguSuriya 7 Mean @ThiyaguSuriya 8 Mean The mean represents the average value of the dataset. It can be calculated as the sum of all the values in the dataset divided by the number of values. @ThiyaguSuriya 9 Mean Symbol (X Bar) Mean is the average of the given numbers and is calculated by dividing the sum of given numbers by the total number of numbers. Mean = ( Sum of all the observations / Total number of observations ) X = (Sum of values ÷ Number of values) X= (x1 + x2 + x3 +….+xn)/n @ThiyaguSuriya 10 Example What is the mean of 2, 4, 6, 8 and 10? Solution: First, add all the numbers. 2 + 4 + 6 + 8 + 10 = 30 Now divide by 5 (total number of observations). Mean = 30/5 = 6 @ThiyaguSuriya 11 Example - Mean X: 8, 6, 7, 11, 3 Sum = 35 N=5 M = 35/5 = 7 @ThiyaguSuriya 12 Example - Mean X X + X + X +... + X å = X= 1 2 3 n n n 57 + 86 + 42 + 38 + 90 + 66 = 6 379 = 6 = 63.167 @ThiyaguSuriya 13 Let's find Ahalya’s MEAN science test score. 783 ÷ 9 + The mean is 87 97 84 88 100 95 63 73 86 97 Mean = ( Sum of all the observations / Total number of observations ) 783 @ThiyaguSuriya 14 Median @ThiyaguSuriya 15 Median The median is the middle value of the dataset in which the dataset is arranged in ascending order or in descending order. The median is the middle score for a data set arranged in order of magnitude. @ThiyaguSuriya 16 Finding the Median 1. Arrange the scores in ascending or descending numerical order 2. Calculate the value of [(N+1)/2] 3. round the {(N+1)/2]th item @ThiyaguSuriya 17 @ThiyaguSuriya 18 Odd number of Observations Even number of Observations @ThiyaguSuriya 19 @ThiyaguSuriya 20 Example - Median X: 6, 6, 7, 8, 9, 10, 11 Median = 8 Y: 1, 3, 5, 6, 8, 12 Median = 5.5 @ThiyaguSuriya 21 63 73 84 86 88 95 97 97 100 The median is 88. Half the numbers are less than the median. Half the numbers are greater than the median. @ThiyaguSuriya 22 Mode @ThiyaguSuriya 23 Mode The mode represents the frequently occurring value in the dataset. Sometimes the dataset may contain multiple modes & in some cases, it does not contain any mode at all. @ThiyaguSuriya The mode is the most frequent score in our data set. 24 @ThiyaguSuriya 25 Mode Score or qualitative category that occurs with the greatest frequency Always used with nominal data, we find the most frequently occurring category Bimodal -- Data sets that have two modes Multimodal -- Data sets that contain more than two modes @ThiyaguSuriya 26 Example - Mode X: 8, 6, 7, 9, 10, 6 Mode = 6 Y: 1, 8, 12, 3, 8, 5, 6 Mode = 8 Can have more than one mode Z: 1, 2, 2, 8, 10, 5, 5, 6 Mode = 2 and 5 @ThiyaguSuriya 27 63 73 84 86 88 95 97 97 100 The value 97 appears twice. All other numbers appear just once. 97 is the MODE @ThiyaguSuriya 28 A Hint for remembering the MODE… The first two letters give you a hint… MOde Most Often @ThiyaguSuriya 29 Which set of data has ONE MODE? A 9, 11, 16, 6, 7, 17, 18 B 18, 7, 10, 7, 18 C 9, 11, 16, 8, 16 @ThiyaguSuriya 30 Which set of data has NO MODE? A 9, 11, 16, 6, 7, 17, 18 B 18, 7, 10, 7, 18 C 13, 12, 12, 11, 12 @ThiyaguSuriya 31 Which set of data has MORE THAN ONE MODE? A 9, 11, 16, 8, 16 B 9, 11, 16, 6, 7, 17, 18 C 18, 7, 10, 7, 18 @ThiyaguSuriya 32 GROUPED DATA @ThiyaguSuriya 33 Mean x = f1x1 + f2x2 + …. + fnxn/f1 + f2+… + fn Mean = ∑(f i.xi)/∑f i Mean = ∑(f.x)/∑N @ThiyaguSuriya 34 S Mean Midpoint (X) CI f fX 95.5 91-100 5 477.5 85.5 81-90 10 855 75.5 71-80 15 1132.5 65.5 61-70 10 655 55.5 51-60 6 333 45.5 41-50 3 136.5 35.5 31-40 1 35.5 N = 50 fX =3625 @ThiyaguSuriya SfX Mean = N M = 3625/50 = 72.5 35 Direct Method for Calculating Mean Step 1: For each class, find the Midpoint / class mark xi, as x=1/2(lower limit + upper limit) Step 2: Calculate f i.xi for each i. Step 3: Use the formula Mean = ∑(f i.xi)/∑f i. Example: Find the mean of the following data Class Interval 0-10 10-20 20-30 30-40 40-50 Frequency 12 16 6 7 9 @ThiyaguSuriya 36 Mean Class Frequency Interval fi Class Mark xi ( f i.xi ) 0-10 12 5 60 10-20 16 15 240 20-30 6 25 150 30-40 7 35 245 40-50 9 45 405 ∑f i=50 SfX Mean = N ∑f i.xi=1100 Mean = ∑(f i.xi)/∑f i = 1100/50 = 22 @ThiyaguSuriya 37 @ThiyaguSuriya 38 Merits of Arithmetic Mean Simple to understand Easy to compute, Capable of further mathematical treatment, Calculated based on all the items of the series, It gives the value which balances the either side, It can be calculated even if some values of the series are missing. It is least affected by fluctuations in sampling. @ThiyaguSuriya 39 Demerits of Arithmetic Mean Extreme Items Have Disproportionate Effect. When Data is Vast, The Calculations Become Tedious. In the case of Openended Classes, the mean can only be calculated by making some assumptions. @ThiyaguSuriya IT Is Not Representative If Series Is Asymmetrical. 40 Median @ThiyaguSuriya 41 Median CCI / ECI CI f CF 90.5 - 100.5 91-100 5 50 80.5 – 90.5 81-90 10 45 Locate the Median Class = N / 2 = 50 / 2 = 25 70.5 – 80.5 71-80 15 35 60.5 – 70.5 61-70 10 20 50.5 – 60.5 51-60 6 10 40.5 – 50.5 41-50 3 4 30.5 – 40.5 31-40 1 1 N = 50 @ThiyaguSuriya 42 MEDIAN ECI/ CCI CI f cf 55.5-60.5 56-60 6 60 50.5-55.5 51-55 9 54 45.5-50.5 46-50 15 45 40.5 (L)-45.5 41-45 13 (f ) 30 35.5-40.5 36-40 10 17 (M) 30.5-35.5 31-35 7 7 LOCATION OF THE MEDIAN CLASS MEDIAN= = L+ 60 = 30 2 ( N 2 - m) ´c f = 40.5 + (60 2 - 17) ´5 13 N = 60 @ThiyaguSuriya 43 Merits of Median Easy to calculate, Can be calculated even if the data is incomplete, It is unaffected in case of asymmetrical series, Useful in case the series of qualitative characteristics is given for example beauty, intelligence etc. Median is a reliable measure of central tendency if in a series, frequencies do not tend to be evenly distributed. Median can be expressed graphically. @ThiyaguSuriya 44 Mode Mo = xk + h{(f k – f k-1)/(2f k – f k-1 – f k+1)} Where, • xk = lower limit of the modal class interval. • fk = frequency of the modal class. • fk-1= frequency of the class preceding the modal class. • fk+1 = frequency of the class succeeding the modal class. • h = width of the class interval. @ThiyaguSuriya 45 Example : Calculate the mode for the following frequency distribution. Class 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Frequency 5 8 7 12 28 20 10 10 Class 40-50 has the maximum frequency, which is called the modal class. xk = 40, h = 10, f k = 28, f k-1 = 12, f k+1 = 20 Mode, Mo= xk + h{(f k – f k-1)/(2f k – f k-1 – f k+1)} = 40 + 10{(28 – 12)/(2 × 28 – 12 – 20)} = 46.67 Hence, mode = 46.67 @ThiyaguSuriya 46 Relationship among mean, median and mode, Mode = 3(Median) – 2(Mean) @ThiyaguSuriya 47 Type of Variable Best measure of central tendency Nominal Mode Ordinal Median Interval/Ratio (not skewed) Mean Interval/Ratio (skewed) Median @ThiyaguSuriya 48 Thank You @ThiyaguSuriya 49