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,