Measures of central tendency and dispersion

Indramani TripathiM.Sc-Biochemistry

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MEASURE OF CENTRAL VALUE

1-CROXTON AND COWDEN-“Average is single value within the the range of data that used to represented all the value in series””2-CLARK &SCHAKADA-“Average is an attempt to find one single to describe whole group of figure”3-Dr B0WELY“Statistics is the science of average”

Since an average is somewhere within the range of data it is also called a Measure of central value

Central tendency = means – average

Characteristics of Average

• ClearDefinationThe average should be properly defined• Easy to calculationIt should have easy to calculation• Representative of series The should be based on each and every item of the seris • UnaffectedThe average should not be affected to the extreme value • Algebraic and mathematical calculation It should be easily subjected to further mathematical calculation

• Absolut number The average should be in Absolut no and the not percentage • SamplingThe average should be least affect by the function of sampling

Limitation of average

• The value of central tendency or average does not completely describe the data

• Average is obtain very judiciously and is ideal for a particular investigation

the best average has its own series• Other limitation of average

• Types of average1. Mathematical Average2. Positional average 3. Partition average

Mathematical average or (mean)

Average represent mathematically is termed as mean

Mean 4. Arithmetic mean5. Geometric mean6. Harmonic mean

Arithmetic mean Arithmetic average mean is the quantity obtained by the sum of all the The value of the items in a series by their number

Formula

Where AM Arethemathic mean =m n=no of item

Formula: Arithmetic Mean = sum of elements / number of elements                = a1 + a2 + a3 + ... + an / n Example to find the Arithmetic Mean of 3, 5, 7. Step 1: The sum of the numbers are:              3 + 5 + 7 = 15 Step 2: The total numbers are:              There are 3 numbers. Step 3: The Arithmetic Mean is:              15 / 3 = 5

Geometric Mean

The geometric mean is defined as the nth root of theProduct of nth observationIt is denoted the GM

Formula

Geometric Mean :Geometric Mean = ((X1)(X2)(X3)........(XN))1/N

whereX = Individual score N = Sample size (Number of scores)

Formula: Geometric Mean = ((x1)(x2)(x3) ... (xn))1/n

 where x = Individual score and n = Sample size (Number of scores) Example to find the Geometric Mean of 1, 2 ,3 ,4 ,5. Step 1: n = 5 is the total number of values. Find 1 / n.              1 / n = 0.2 Step 2: Find Geometric Mean using the formula:              [(1)(2)(3)(4)(5)]0.2 = 1200.2

              Geometric Mean = 2.60517

Harmonic Mean

The harmonic mean is defined as the

“The reciprocal of the arithmetic mean of the value reciprocal of given value”

Formula

Harmonic Mean Formula :Harmonic Mean = N/(1/a1+1/a2+1/a3+1/a4+.......+1/aN) Where, X = Individual score N = Sample size (Number of scores)

Harmonic Mean

Example:

To find the Harmonic Mean of 1,2,3,4,5.Step 1Calculate the total number of values. N = 5Step 2:Now find Harmonic Mean using the above formula. N/(1/a1+1/a2+1/a3+1/a4+.......+1/aN) = 5/(1/1+1/2+1/3+1/4+1/5) = 5/(1+0.5+0.33+0.25+0.2) = 5/2.28 So Harmonic Mean = 2.19

Average of position

Averages of the postion indicate the postion of the an average in series of observation arranged in inccrasing of the magnitude

Average of the position

1. Median 2. Mode

Median=Median means value of the middle most observation when data Arranged ascending or desending order of magnitude

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Example 2:To find the median of 4,5,7,2,1,8 [Even]Step 1:Count the total numbers given. There are 6 elements or numbers in the distribution.Step 2:Arrange the numbers in ascending order. 1,2,4,5,7,8Step 3:The total elements in the distribution (6) is even. As the total is even, we have to take average of number at n/2 and (n/2)+1 So the position are n/2= 6/2 = 3 and 4 The number at 3rd and 4th position are 4,5Step 4:Find the median. The average is (4+5)/2 = Median = 4.5

ModeThat value of the variable for which the frequency is maximum It is denoted by the MO

• Unimodal frequency distribution Data having one mode• Biomodal frequency distribution Data having two mode• Multimodal frequency distribution Data having two or more mode • AntimodeIn u shaped distribution the law point at the middle of the distribution

Example

The following is the number of problems that Ms. Matty assigned for homework on 10 different days. What is the mode?8, 11, 9, 14, 9, 15, 18, 6, 9, 10Solution:Ordering the data from least to greatest, we get:6, 8, 9, 9, 9, 10, 11 14, 15, 18Answer:The mode is 9.

Statistical dispersionn

In statistics, dispersion (also called variability, scatter, or spread) denotes how stretched or squeezed a distribution (theoretical or that underlying a statistical sample) is. Common examples of measures of statistical dispersion are the variance, standard deviation and interquartile range.

Types of Measures of Dispersions:

Measures of dispersion are of two types   (i )  Measures of Absolute Dispersion, and  (ii)    Measures of Relative Dispersion.

(i) Measures of Absolute Dispersion: The actual variation or dispersion determined by the Measures of Absolute Dispersion is called ‘absolute dispersion’.

(ii) Measures of Relative Dispersion: The measures of absolute dispersion cannot be used to compare the variation of two or more series. For

To compare the variation of two or more series, we need a measure of relative dispersion. It is defined as: