Basic of Bio-statistics measure of central tendency

1. Central Tendency
Measure of Central Tendency
 Measures of central tendency are a combination of two words i.e. ‘Measure’ and
‘Central tendency’. Measure means methods and central tendency means average
value of any statistical series and Central Tendency means the methods of finding out
the central value or average value of a statistical series of quantitative information.
 In other words, Central Tendency may be defined as value of the variate which is
thoroughly representative of the series or the distribution as a whole.
 The measure of this character of an average depends upon how closely the other
observation cling to it.
Uses of Central Tendency
 Average provides the overall picture of the series. We cannot remember each and
every facts relating to a field of enquiry.
 Average value provides a clear picture about the field under study for guidance and
necessary conclusion.
 It gives a concise description of the performance of the group as a whole and it enables
us to compare two or more groups in terms of typical performance.
Characteristics of Central Tendency or Average
1) It should be rigidly defined and not left to be estimated by the investigator.
2) Its computation should be based on all the observation.
3) The general nature of an average should be easily comprehensible. It should not be of
too abstract a mathematical characters.
4) An average should be capable of being computed easily and rapidly.
5) An average should be as little affected by the fluctuation of sampling as possible.
6) An average should be able to lend itself readily to algebraic treatment.
Types of central tendency
a) Arithmetic mean
b) Median
c) Mode
d) Geometric mean
e) Harmonic mean
f) Weighted mean
Arithmetic mean
An arithmetic average is defined as the “quotient obtained by dividing the total of the values
of a variable by the total number of their observations or items.”

2. The formula for calculate mean from un-grouped data is:
From the grouped data the mean is calculated by the following formula:
Uses of Arithmetic Mean
 Mean is the centre of gravity in the distribution and each score contributes to the
determination of it when the spread of the scores are symmetrically around a central
point.
 Mean is more stable than the median and mode. So that when the measure of central
tendency having the greatest stability is wanted mean is used.
 Mean is used to calculate other statistics like S.D., coefficient of correlation, ANOVA,
ANCOVA etc.
Merits of Arithmetic Mean
1) It is the most popular central tendency as it is easy to understand.
2) It is easy to calculate.
3) It includes all the scores of a distribution.
4) It is not affected by sampling so that the result is reliable.
5) Mean is capable of further algebraic treatment so that different other statistics like
dispersion, correlation, skew-ness requires mean for calculation.
Demerits of Arithmetic Mean
1) Mean is affected by extreme scores.
2) Sometimes mean is a value which is not present in the series.
3) Sometimes it gives absurd values. For example, there are 41, 44 and 42 students in
class VIII, IX and X of a school. So the average students per class are 42.33. It is never
possible.
4) In case of open ended class intervals, it cannot be calculated without assuming the
size of the open end classes.
Median
it is defined as “the middle value in a distribution, below and above which lie values with
equal total frequencies or probabilities.”

3. In other words, when all the observations are arranged in ascending or descending order of
magnitude, the middle is known as the median.
Calculation of Median
Uses of Median
1) Median is used when the exact midpoint of the distribution is needed or the 50% point
is wanted.
2) When extreme scores affect the mean at that time median is the best measure of
central tendency.
3) Median is used when it is required that certain scores should affect the central
tendency, but all that is known about them is that they are above or below the median.
4) Median is used when the classes are open ended or it is of un equal cell size.
Merits of Median:
1) It is easy to compute and understand.
2) All the observations are not required for its computation.
3) Extreme scores does not affect the median.
4) It can be determined from open ended series.
5) It can be determined from un-equal class intervals.
Demerits of Median:
1) It is not rigidly defined like mean because its value cannot be computed but located.
2) It does not include all the observations.
3) It cannot be further treated algebraically like mean.
4) It requires arrangement of the scores or class intervals in ascending or descending
order.
5) Sometimes it produces a value which is not found in the series.
Mode
This is that value of the variable which occurs most frequently or whose frequency is
maximum.
Also, if several samples are drawn from a population, the important value which appear
repeatedly in all the sample is called the mode.

4. The mode is 6
(it occurs most often)
Uses of mode
1) When we want a quick and approximate measure of central tendency.
2) When we want a measure of central tendency which should be typical value.
Merits of Mode
1) Mode gives the most representative value of a series.
2) Mode is not affected by any extreme scores like mean.
3) It can be determined from an open ended class interval.
4) It helps in analysing qualitative data.
5) Mode can also be determined graphically through histogram or frequency polygon.
6) Mode is easy to understand.
Demerits of Mode
1) Mode is not defined rigidly like mean. In certain cases, it may come out with different
results.
2) It does not include all the observations of a distribution but on the concentration of
frequencies of the items.
3) Further algebraic treatment cannot be done with mode like mean.
4) In multimodal and bimodal cases, it is difficult to determine.
5) Mode cannot be determined from unequal class intervals.
6) There are different methods and different formulae which yield different results of
mode and so it is rightly remarked as the most ill-defined average.
Geometric mean
Arithmetic mean, although gives equal weightage to all the items, has got a tendency towards
the higher values. Sometimes we want an average having a tendency towards the lower
values so in such case we take the help of geometric mean.
Formula
Its Computation is not possible without the help of logarithm. Therefore, using logarithm,
log(G.M.) = (fllogX1 + f2logX2 +………. +fk logXk) =

5. Uses of Geometrical Mean
1) The geometric mean is most useful when numbers in the series are not independent
of each other or if numbers tend to make large fluctuations.
2) Applications of the geometric mean are most common in business and finance, where
it is commonly used when dealing with percentages to calculate growth rates and
returns.
3) It is also used in certain financial and stock market indexes, such as Financial Times'
Value Line Geometric index.
Merits of Geometric Mean
1) It is based on all the items of the data.
2) It is rigidly defined. It means different investigators will find the same result from the
given set of data.
3) It is a relative measure and given less importance to large items and more to small
ones unlike the arithmetic mean.
4) Geometric mean is useful in ratios and percentages and in determining rates of
increase or decrease.
5) It is capable of algebraic treatment. It means we can find out the combined geometric
mean of two or more series.
Demerits of Geometric Mean
1) It is not easily understood and therefore is not widely used.
2) It is difficult to compute as it involves the knowledge of ratios, roots, logs and antilog.
3) It becomes indeterminate in case any value in the given series happen to be zero or
negative.
4) With open-end class intervals of the data, geometric mean cannot be calculated.
5) Geometric mean may not correspond to any value of the given data.
Harmonic Mean
Harmonic mean of a number of quantities is the reciprocal of the arithmetic mean of their
reciprocal.
Formula
Uses of Harmonic Mean
The harmonic Mean is restricted in its field of application. It is useful in computing the average
rate of increasing of profits or average speed at which a journey has been performed or
average price at which an article has been sold.

6. Merits of Harmonic Mean
1) Like AM and GM, it is also based on all observations.
2) It is most appropriate average under conditions of wide variations among the items of
a series since it gives larger weight to smaller items.
3) It is capable of further algebraic treatment.
4) It is extremely useful while averaging certain types of rates and ratios.
Demerits of Harmonic Mean
1) It is difficult to understand and to compute.
2) It cannot be computed when one of the values is 0 or negative.
3) It is necessary to know all the items of a series before it can be calculated.
4) It is usually a value which may not be a member of the given set of numbers.
Weighted mean
 In the calculation of the arithmetic mean every item is given equal importance or is
equally weighted. But sometimes it so happens that all the items are not equal
importance.
 At that time they are given proper weights according to their relative importance, and
then the average which is calculated on the basis of these weights is called the
weighted average or weighted mean.
Formula
Uses of Weighted Mean
1) When the number of individuals in different classes of a group are widely varying.
2) When the importance of all the items in a series is not the same.
3) When the ratios, percentages or rates (e.g. quintals per hectare, rupees per kilogram,
or rupees per meter etc.) are to be averaged.
4) When the means of a series or group is to be obtained from the means of its
component parts.
5) Weighted mean is particularly used in calculating birth rates, death rates, index
numbers, average yield, etc.
Example:
The data on the length (mm) of 20 types of wools are given below.
Find the Arithmetic Mean, Geometric Mean, Harmonic Mean, Median and Mode.
138, 138, 132, 149,164, 146, 147, 152,115, 168,
176, 154,132, 146, 147, 140,144, 161, 142, 145

7. Answer:
Step 1: Formulate a table of following type
Arithmetic Mean(AM):
Geometric Mean:
Harmonic Mean:

8. Median:
The data in the ascending order of magnitude is
132,132,138,138,140,142,144,145,146,146,147,
147,149,150,152,154,161,164,168,176
Mode:
The given set of data is polymodal type. 132,138,146 and 147 are Repeated twice.
Hence there are four modes.:
Mode: 132, 138, 146, 147

9. Measures of Dispersion
 It is quite obvious that for studying a series, a study of the extent of scatter of the
observation of dispersion is also essential along with the study of the central tendency
in order throw more light on the nature of the series.
 Simply dispersion (also called variability, scatter, or spread) is the extent to which
a distribution is stretched or squeezed.
Different measures of dispersion
1) Range
2) Mean deviation
3) Standard deviation
4) Variance
5) Quartile Deviation
6) Coefficient of Variation
7) Standard Error
Range
Range is the simplest measure of dispersion.
It is the difference the between highest and the lowest terms of a series of observations.
Range = XH – XL
where, XH = Highest variate value
XL = Lowest variate value
Properties
 Its value usually increases with the increase in the size of the sample.
 It is usually unstable in repeated sampling experiments of the same size and large
ones.
 It is very rough measure of dispersion and is entirely unsuitable for precise and
accurate studies.
 The only merits possessed by ‘Range’ are that it is (i) simple, (ii) easy to understand
(iii) quickly calculated.
Mean deviation
 The deviation without any plus or minus sign are known as absolute deviations.
 The mean of these absolute deviations is called the mean deviation.
 If the deviations are calculated from the mean, the measure of dispersion is called
mean deviation about the mean.

10. Characteristics of mean Deviation
 A notable characteristic of mean deviation is that it is the least when calculated about
the Median .
 Standard deviation is not less than the mean deviation in a discrete, i.e., it is either to
or greater than the M.D. about Mean
 When a greater accuracy is required, standard deviation is used as a measure of
dispersion.
 When an average other than the A.M. Is calculated as a measure of central tendency
M. D. about that average is the only suitable measure of dispersion.
Standard deviation
Its calculation is also based on the deviations from the arithmetic mean. In case of mean
deviation, the difficulty, that the sum of the deviations from the arithmetic mean is always
zero, is solved by taking these deviation irrespective of plus or minus signs. But here, that
difficulty is solved by squaring them and taking the square root of their average.
It is thus defined by the following expression.
Standard Deviation (S. D.):
Where, X = An observation or variate value
µ = Arithmetic mean of the population
N = Number of given observations
Characteristics and uses of S.D.
 It is rigidly defined.
 Its computation is based on all the observation.
 If all the variate values are the same, S.D.=0

11.  S.D. is least affected by fluctuations of sampling.
 It is used in computing different statistical quantities like, regression coefficients,
correlation coefficient, etc.
Variance
Variance is the square of the standard deviation.
Variance= (S. D.)2
This term is now being used very extensively in the statistical analysis of the results from
experiments.
The variance of a population is generally represented by the symbol σ² and its unbiased
estimate calculated from the sample, by the symbol s².
Quartile Deviation or Semi-Inter-Quartile Range.
This measure of dispersion is expressed in terms of quartiles and known quartile deviation or
semi-inter-quartile range.
where, Q1=Lower Quartile
Q3=Lower Quartile
It is not a measure of the deviation from any particular average. For symmetrical and
moderately skew distribution the quartile deviation is usually two-third of the standard
deviation.
Coefficient of Variation
This is also a relative measure of dispersion, and it is especially important on account of the
the widely used measure of central tendency and dispersion i.e., Arithmetic Mean and
Standard deviation.
It is given by
It is expressed in percentage, and used to compare the variability in the two or more series.
Standard Error
 The term ‘Standard error’ of any estimate is used for a measure of the average
magnitude of the difference between the sample estimate and the population
parameter taken over all possible samples of the same size, from the population.
 This term is applied for the standard deviation of the sampling distribution of any
estimate.

12.  If S be the standard deviation of the sample size N, the estimate of the standard error
of mean is given by
Standard Error:
Example 1
Find Range, Quartile Deviation, Mean Deviation about x and Standard Deviation and their
relative measure for the following data:
1.3, 1.1, 1.0, 2.0, 1.7, 2.0, 1.9, 1.8, 1.6, 1.5
ANSWER:
On arranging the above data in ascending order, we get
1.0, 1.1, 1.3, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.0
Range: H-L= 2.0-1.0 = 1.0
Quartile Deviation:
Mean Deviation about Mean

13. Formulate table of this kind.
Mean Deviation
Standard Deviation:

14. Co-efficient variation:
Example 2
Calculate Range, Quartile Deviation, Mean Deviation about Mean, Median and Mode,
Standard Deviation and their relative measures for the following data.

15. Answer: Formulate a table of this kind
Range:
Quartile Deviation:

16. Mean Deviation:
Standard Deviation and Co-efficient of Variation:

17. Normal Distribution
The equation of the normal curve is
Normal Distribution Curve
Properties of a Normal Distribution
 The distribution curve is symmetrical about thr mean µ and falls rapidly on both the
side tailing off asymptotically to the X-axis in both direction(i.e., the X-axis is tangent
to the curve at infinity).
 There are only two independent parameter, µ and σ.
 Here, Mean = Median = Mode= µ
 The first and third moment about the mean are zero i.e., µ1 = 0 and µ3 = 0.
 The second moment about the mean is σ2, the variance of the distribution, µ2 = σ2.
 The fourth moment about the mean is 3σ4 i.e., µ4 = 3σ4.
 In the normal distribution, β1 = 0 and β2 = 3.

18. (i) The range µ ± σ includes about the 68% of the observation.
(ii) The range µ ± 2σ includes about the 95% of the observation.
(iii) The range µ ± 3σ includes about the 99% of the observation.
 A remarkable property of normal distribution is that sums and difference of normally
distributed variables are also normally distributed.