Measures Of Central Tendency PDF

Measures of Central Tendency Mr. Bashir Ullah MS Epi & biostatistics KMU_IPHSS BASHIR 1 1/17/2025  0bjectives By the end of this lecture...

Measures of Central Tendency Mr. Bashir Ullah MS Epi & biostatistics KMU_IPHSS BASHIR 1 1/17/2025  0bjectives By the end of this lecture the students will be able to: 1. compute and distinguished between the uses of Measures of Central Tendency: Arithmetic mean , Median , Mode. 2. understand the distinction between population Mean and Sample Mean. BASHIR 2 1/17/2025  Measures Of Central Tendency  A data set can be summarized into a single value, usually lies somewhere in the center and represent the whole data set.  Such a single value that represent the central part of a data set is called central value.  Tendency of observations that cluster in the central part of the data set is called central tendency. “OR”  Several statistics that can be used to represent the “center” of the distribution.  These statistics are called measures of central tendency. BASHIR 3 1/17/2025  Most Commonly Used Measure Of Central Tendency Measure of Central Tendency Arithmetic Median Mode Mean BASHIR 4 1/17/2025  1.Arithmetic Mean  Simply it is called mean or average and mostly used measure of central tendency in every field of research.  “Arithmetic mean is a value obtained by dividing the sum of all observations in a data set by the number of observations”.  The values of the data are represented by X’s. BASHIR 5 1/17/2025  Mathematical Description of Arithmetic Mean  Mathematically, Arithmetic mean is expressed as N x  x2   xN x i  = 1 = i 1 [Population data] N N n x  x2   xn x i 1 i X = 1 = [Sample data] n n BASHIR 6 1/17/2025  Example  Let us compute the mean (or average) of this sample:    110  118  110  122  110  150  120 = n 6  In the above example, there are some new mathematical notations. BASHIR 7 1/17/2025  Example (continue)    110  118  110  122  110  150 = n 6  120  First, a symbol that denotes the mean of the sample. BASHIR 8 1/17/2025  Example (continue) The second part of the equation shows how this quantity is computed or the formula BASHIR 9 1/17/2025  Example (continue) The sigma symbol summation (  ) tells us to sum all the individual Xs. BASHIR 10 1/17/2025  Example (continue)  Lastly, we must divide by ‘n’, that is: the number of observations. BASHIR 11 1/17/2025  Example  The data show the number of patients in a sample of six hospitals who acquired an infection while hospitalized. 110 76 29 38 105 31  Find the mean.  Solution:  The mean of the number of hospital infections for the six hospitals is 64.8. BASHIR 12 1/17/2025  Arithmetic Mean for Grouped Data  Mathematically, Arithmetic mean for grouped data (frequency distribution ) is expressed as f1 x1  f 2 x2   f n xn X = f1  f 2   fn n fx i 1 i i = f i i   fx , where  f  n (total number of observations)  f where, f1 , f 2 , , f n are corresponding frequencies BASHIR 13 1/17/2025 of x1 , x2 , , xn  Example-1  Consider the following frequency distribution of the salaries of 50 employees of a certain University , compute arithmetic mean. Salary (000)[x] 40 50 60 70 80 90 Total Number of employees [f] 20 10 8 5 4 3 50 fx 800 500 480 350 320 270 2720 n  f x i i 2720 X = i 1   54.4  f i 50 BASHIR 14 1/17/2025 It shows that each employee of the University has 54.4 (thousand) salary, on the average.  Finding the Mean of Group Data BASHIR 15 1/17/2025  Example-2 The data represent the number of miles run during one week for a sample of 20 runners. Solution :  Step 1: Make a table as shown. BASHIR 16 1/17/2025  Example-2 (continue)  step 2: Find midpoint of each class and enter into column C.  Step 3: For each class, multiply the frequency by the midpoint, as shown, and place the product in column D. BASHIR 17 1/17/2025  Example-2 (continue) COMPLETE TABLE AS SHOWN HERE Step 4: Find the sum of column D. Step 5 : Divide the sum by n to get the mean. BASHIR 18 1/17/2025  Properties of the Mean 1. Uniqueness. For a given set of data there is one and only one arithmetic mean. 2. Simplicity. The arithmetic mean is easily understood and easy to compute. 3. Since each and every value in a set of data enters into the computation of the mean, it is affected by each value. Extreme values, therefore, have an influence on the mean and, in some cases, can so distort it that it becomes undesirable as a measure of central tendency. BASHIR 19 1/17/2025  DEFINITIONS Class Boundaries: In a grouped frequency distribution, if upper limit of a class is repeated as a lower limit of the next class, such classes are called class boundaries. For example, the classes 5-10, 10-15, 15-20, 20-25, 25-30 and 30-35 are the class boundaries. Class Limits : In a grouped frequency distribution, if upper limit of a class is not repeated as a lower limit of the next class such as, 5-9, 10-14, 15-19, 20-24, 25-29, 30-34 and 35-39 are called class limits. BASHIR 20 1/17/2025  Creating Class boundaries These class limits can be converted in to class boundaries by taking the following steps: 1. Compute the mid-way value = (Lower limit of the 2nd class – upper limit of 1st class)/2 =(10-9)/2 = 0.5 2. Now subtract mid-way value from each of the lower class limit and add it with the upper class limit. In such a case the class/groups so formed are become class boundaries. BASHIR 21 1/17/2025  Example Creating Class bound CLASS LIMITS MID WAY VALUE MID WAY VALUE CLASS SUBTRACT FROM BOUNDARIES LOWER LIMIT AND ADD TO UPPER LIMIT OF EACH CLASS INTERVAL 5___9 10-9/2= 0.5 5-(0.5)___9+(0.5) 4.5___9.5 10___14 15-14/2=0.5 10-(0.5)___14+(0.5) 9.5___14.5 15___19 20-19/2=0.5 15-(0.5)___19+(0.5) 14.5___19.5 20___24 25-24/2=0.5 20-(0.5)___24+(0.5) 19.5___24.5 25___29 30-29/2=0.5 25-(0.5)___29+(0.5) 24.5___29.5 30___34 35-34/2=0.5 30-(0.5)___34+(0.5) 29.5___34.5 35__39 =0.5 35-(0.5)____39+( 0.5) 34.5___39.5 BASHIR 22 1/17/2025  2-Median  Median is a value which divide and arranged data set into two equal parts i.e. half (50%) of the observations will lies below and half (50%) will come above that value. “OR”  Median is the middle value of the measurement when they are arranged in ascending or descending order. Odd Numbers:  If the number of values i-e n is an odd number, the 𝑵+𝟏 median is calculated by Median: The value of( )th 𝟐 item. BASHIR 23 1/17/2025  Odd Number  E.g. Age of 5 students. Unarranged – 1. 5. 3. 2. 4 Ascending order – 1, 2, 3, 4, 5 So 5+1/2 = 6/2= 3 (so value of 3rd item is median) Median in this case is 3.  Similarly, for the data set having the size (even number) divisible by 2, median will be the average of two middle values, for example: 9, 10, 12, 13, 14, 15, 16, 20 (here n = 8) so Median = (13+14)/2 = 13.5 BASHIR 24 1/17/2025  Even Number example: 9, 10, 12, 13, 14, 15, 16, 20 (here n = 8) 8th 8th +2 so the value of ½ ( ) item and value of ( 2 ) item 2 Here 4th and 5th item is 13 and 14 respectively ½ (13+14) = 13.5 BASHIR 25 1/17/2025  Median for Group Data Continuous series BASHIR 26 1/17/2025  Example Step1: Cumulative Frequency Step2: Find Median Class Step5: Frequency of Step3: Find LCL of median class median class Step6: CF of the class Step4: width of the proceeding the median median class class BASHIR 27 1/17/2025  Median for Group Data Discrete Series  In discrete series, the 𝑛+1 size of( ) th item 2 is taken is median. BASHIR 28 1/17/2025  3-MODE  Mode is a value which has maximum frequency as compared to other items of a data set. OR, the most frequent value of a data set is called mode.  A distribution/data set having only one mode is called uni-modal distribution. Similarly, a distribution is defined to be bi-modal if it has two modes. Generally, a distribution having more than one modes is called multi-modal distribution.  If all the observations of a data set have the same frequencies (repeated the same number of times), the data set will have no mode. For example: 2, 4, 6, 4, 8, 10, 8, 10, 6, 2: this data set has no mode because each and every observation is repeated the same number of BASHIR times. 29 1/17/2025  EXAMPLES MODE a) Observation occurring most frequently. 30, 32, 35, 37, 41, 45, 41 mode = 41 years b) A data may have more then one mode 30, 32, 35, 37, 41, 45, 41, 32 mode = 32 & 41 BASHIR 30 1/17/2025  Uni-modal distribution Bi-modal distribution 60 60 Number of students Number of students 45 45 30 30 15 15 0 0 2.7 3.2 3.7 4.2 2.7 3.2 3.7 4.2 GPA GPA 60 Number of students 45 30 Tri-modal distribution 15 0 2.7 3 3.3 3.6 3.9 4.2 BASHIR 31 1/17/2025 GPA  Advantages and Disadvantages of each measurement Advantages Disadvantages Mean 1. computation is easy 1. give impossible figure in discrete 2. result is firm, established 2. affected by extreme values 3. uses all the data 3. needs all the data for calculation. 4. Can be used in statistical work Median 1. not affected by extreme values 1. cannot be used in further statistical 2. exact value can be computed work. 3. of immense importance in 2. it needs order to data. experimental studies 3. does not comprise of all data Mode 1. not affected by extreme 1. there may be more then one mode values 2. does not represent all data 2. histograms can be obtained 3. not used in further statistical work. BASHIR 3. easy to find & understand 32 1/17/2025 BASHIR 33 1/17/2025 BASHIR 34 1/17/2025