4 central tendency

Measures of central Tendency

Dept of AGB Veterinary College, Hebbal

Definition Measure of central tendency are typical values around which other values of distribution concentrate.

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Basic measures of Central tendency • • • • •

Arithmetic mean Median Mode Geometric mean Harmonic mean

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

The Mean • Definition: The arithmetic average obtained by adding up all the scores and dividing by the total number of scores • Most commonly used measure of central tendency • Appropriate for variables that are intervalratio

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Calculating the mean

X 

X i  N

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Mean Example Scores on a exam 68 90 77 79 80 92

X  X 

68  90  77  79  80  92 X  6 486 X  6 X  81 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

i

N

Mean – the average of a group of numbers.

2, 5, 2, 1, 5 Mean = 3 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Mean is found by evening out the numbers

2, 5, 2, 1, 5

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Mean is found by evening out the numbers

2, 5, 2, 1, 5 mean = 3

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Uses Arithmetic mean • For distribution in a concise manner • For comparative study of different distributions.

• For comparing various other statistical measures such as dispersion, skewness, kurtosis and other basic characteristics of data.

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Properties of mean • The algebraic sum of deviations of the given set of observations from their arithmetic mean is zero.

• If all the observations of a series are added, subtracted, multiplied or divided by a constant the mean is also added, subtracted, multiplied or divided by the same constant.

• e.g: The mean of 10 observations is 35. If each observation is increased by 5, then the mean is also increased byDr R5Jayashree, i.e 35Asst. + Prof.

5 = 40 (AGB), Veterinary College,

Merits of Arithmetic mean • • • • •

It is rigidly defined It is easy to calculate and understand It based on all observations It is less affected by sampling Further mathematical analysis can be done

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Demerits • It is very much affected by extreme values.

• It cannot be used if we are dealing with qualitative characters.

• In extremely asymmetrical distribution (skewed) usually arithmetic mean is not representative of the distribution.

• It may lead to wrong conclusions if the details are not available.

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Weighted Arithmetic mean • Some items in a distribution may be less important than other item. In such cases proper weightage is to be given to various items.

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Median- Definition • Is that value of the variable which divides the group in two equal parts, one part greater than the median and other part less than the median.

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Definition • Median – the middle number in a set of ordered numbers.

1, 3, 7, 10, 13 Median = 7 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

How to Find the Median in a Group of Numbers • Step 1 – Arrange the numbers in order from least to greatest.

21, 18, 24, 19, 27 18, 19, 21, 24, 27 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

How to Find the Median in a Group of Numbers • Step 2 – Find the middle number.

21, 18, 24, 19, 27 18, 19, 21, 24, 27 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

How to Find the Median in a Group of Numbers • Step 2 – Find the middle number.

18, 19, 21, 24, 27 This is your median number.

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

How to Find the Median in a Group of Numbers • Step 3 – If there are two middle numbers, find the mean of these two numbers.

18, 19, 21, 25, 27, 28 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

How to Find the Median in a Group of Numbers • Step 3 – If there are two middle numbers, find the mean of these two numbers.

21+ 25 = 46 median 23 2) 46 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

What is the median of these numbers?

16, 10, 7 7, 10, 16

10 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

What is the median of these numbers?

29, 8, 4, 11, 19 4, 8, 11, 19, 29

11 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

What is the median of these numbers?

31, 7, 2, 12, 14, 19 2, 7, 12, 14, 19, 31

13

12 + 14 = 26 2) 26 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

What is the median of these numbers?

53, 5, 81, 67, 25, 78 5, 25, 53, 67, 78, 81 60 53 + 67 = 120 2) 120 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Merits of median • It is rigidly defined.

• It is easy to understand and calculate.

• Since it is positional average , it is not affected by extreme values.

• It is the average to be used in qualitative characters.

• It is useful for skewed distribution Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Demerits of median • In case of even number of data median cannot be determined exactly.

• It is not based on all the observations.

• It cannot be used for further mathematical treatment.

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Quartiles • The values which divide the given data into four equal parts are known as quartiles.

• Q1- It is the value which has 25% of the item below and 75% of the items above it.

• Q2- It is the median which has equal number above and below.

• Q3- It has 25% above and 75 % below it.

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Definition

Mode is the most Popular Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Definition • Mode – the most popular or that which is in fashion.

Baseball caps are a la mode today.

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Definition • Mode – the number that appears most frequently in a set of numbers.

1, 1, 3, 7, 10, 13 Mode = 1 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Merits of mode • Easy to understand and calculate. • Not affected by extreme values.

In a symmetric curve, the mean, median and mode are same.

For a asymmetrical distribution mean and mode will on both sides of the median.

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Demerits • It is not rapidly defined.

• It is not based on all observations.

• It is not suitable for further mathematical treatments.

• It is affected by fluctuating samples.

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

• Mode= Mean-3(Mean-median) = 3 median -2 mean • For a positively skewed distribution mean will be greater than median and median will be greater than mode.

M > Md > Mo In a negatively skewed distribution Mo > Md > M Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Geometric mean • It is defined as nth root of the product of the n number of observations ie G.M=N

x , x , x ........x 1

2

3

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

N

Geometric mean 1 Log GM = logx1 + logx2 + ........lo gxN  N ∑logX Log GM = n GM= Antilog of ∑logX

n Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Geometric mean 29, 8, 4, 11, 19 G.M= 5

29 × 8 × 4 × 11× 19

Log GM = ∑logX

n G.M= Antilog∑logx

n

Log G.M = log 29+log8+log4+log11+log19 5 Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Merits • Based on all observations • It can be used for further analysis.

Demerits • When any value is 0 GM also become 0 • Used in averages, ratios and %.

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Harmonic mean • Harmonic mean is the reciprocal of the arithmetic mean of the reciprocal of the given observations

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Merits • It is rigidly defined, based on all observations and is capable of further algebraic treatment.

• It is less affected by extreme values.

Demerits • It is not simple to understand • It is not easy to calculate as it involves reciprocals • Oes not possess sampling ability Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Situations where different averages are used • AM  Applicable for all sorts of data.  Shall be used in symmetrical distribution and further statistical analysis has to computed for comparison • GM Is used when it is desired to give more weights to small items and  less weight to large items  In the case of ratios, percentages and R Jayashree, Asst. Prof. microorganisms Dr(AGB), Veterinary College,

• HM  It is used in averaging certain types of ratios and problems involving time  It gives more weight to small items Median  Used when the attribute of the data are not directly measurable  Used when the distribution is highly skewed or the extreme values distort the mean  Mode- used to know the most common item

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,

Thank you

Dr R Jayashree, Asst. Prof.

(AGB), Veterinary College,