measures of central tendency

2. Data Description
1. Summarize data, using measures of central tendency,
such as the mean, median, mode, and midrange.
2. Describe data, using measures of variation, such as the
range, variance, and standard deviation.
3. Identify the position of a data value in a data set, using
various measures of position, such as percentiles,
deciles, and quartiles.
4. Use the techniques of exploratory data analysis,
including boxplots and five-number summaries, to
discover various aspects of data.

3. Measures of Central Tendency
• A central tendency is a single value which is used to represent an entire set of data.
• All the data values clustered around central value.
• In simple words it is the tendency of the observations(Data Values)to concentrate
around a central point.
• Statistical measures that indicate the location or position of a central value to describe
the central tendency of the entire data are called Measures of Central Tendency.
Some important measures of central tendency are:
• Mean
• Median
• Mode
• Quartiles
• Deciles

4. Characteristics of Measures of Central Tendency
• It should be easy to understand
• It should be easy to compute
• It should be based on all the observations
• It should be rigidly defined i.e. it must have one and only one
interpretation.
• It should be capable of further algebraic treatments i.e. it is used for
further algebraic computations.
• It should have sampling stability i.e. if we take say 10 different
samples form the population it will result into almost same measures
of central values.
• It should not be unduly effected by the presence of extreme values

5. Arithmetic Mean
• The mean, also known as
the arithmetic average, is
found by adding the
values of the data and
dividing by the total
number of values.
For Grouped Data

6. Properties of Arithmetic Mean
1. The sum of the deviations of the items from the arithmetic mean is
always zero i.e 𝑥 − 𝑥 = 0
2. The sum of the squares of the deviations of a set of values is
minimum when taken from mean. i.e. 𝑥 − 𝑥 2 is minimum.
3. Simple arithmetic means may be combined to give composite
mean.
𝑥12 =
𝑁1𝑥1+𝑁2𝑥2
ℕ1+𝑁2

7. Median
• The median of a distribution is the middle of Central value of the variable when
the values are arranged in the order of their magnitude.
• It divides the distribution into two equal parts so that half of the data values less
than the median while the other half of the values greater than the median.
• For ungrouped data:
For odd number of observations, Median = [(n + 1)/2]th term.
For even number of observations, Median = [(n/2)th term + ((n/2) + 1)th term]/2
• For grouped data:
Median = L + [((n/2) - pcf)/f] × i where,
L = Lower limit of the median class
pcf = Preceding Cumulative frequency to median class
i = Class size
n = Number of observations
Median class = Class where n/2th observation lies

8. Mode
• The mode is the most commonly occurring value in a distribution.
For Grouped data :
𝑀𝑜𝑑𝑒 = 𝐿 +
𝑓1−𝑓0
2𝑓1−𝑓0−𝑓2
𝑥 ⅈ where,
L= Lower class interval of Modal class
𝑓1 = Frequency of Modal Class
𝑓0 = Frequency preceding Modal Class
𝑓2 = Frequency Succeeding Modal Class
i = Width of class Interval of Modal Class

9. Properties & Uses of Central Tendency
The Mean
1. The mean is found by using all the values of the data.
2. The mean varies less than the median or mode when samples are taken from the same population and all
three measures are computed for these samples.
3. The mean is used in computing other statistics, such as the variance.
4. The mean for the data set is unique and not necessarily one of the data values.
5. The mean cannot be computed for the data in a frequency distribution that has an open-ended class.
6. The mean is affected by extremely high or low values, called outliers, and may not be the appropriate
average to use in these situations.
The Median
1. The median is used to find the center or middle value of a data set.
2. The median is used when it is necessary to find out whether the data values fall into the upper half or
lower half of the distribution.
3. The median is used for an open-ended distribution.
4. The median is affected less than the mean by extremely high or extremely low values.
The Mode
1. The mode is used when the most typical case is desired.
2. The mode is the easiest average to compute.
3. The mode can be used when the data are nominal or categorical, such as religious preference, gender, or
political affiliation.
4. The mode is not always unique. A data set can have more than one mode, or the mode may not exist for a
data set.

10. Measures of Dispersion/Variation
Dispersion means scatteredness
• The degree to which the numerical data tends to spread around an
average value is called dispersion or variation of data.
• Methods of Studying Dispersion:
1. Range
2. Quartile Deviation
3. Average Deviation
4. Standard Deviation

11. Significance of Measuring variation
• To determine the reliability of an average : Measures of variations tells
whether an average is representative of the entire data or not. If variation
is small then we say that average is representative of the entire data.
• To serve as a basis for the control of the variability : Variation also
determines the nature and cause of variation in order to control the
variation itself. It helps in determining the reason behind variation
• To compare two or more series with regard to their variability: It also
enables to compare two or more series with respect to their variability. The
series with less variation is more uniform and consistent.
• To facilitate the use of other statistical measures: Variation is also used in
other statistical techniques like correlation, testing of hypothesis,
production control, cost control etc.

12. Properties of Measures Variation
• It should be easy to understand
• It should be easy to compute
• It should be based on all the observations
• It should be rigidly defined i.e. it must have one and only one
interpretation.
• It should be capable of further algebraic treatments i.e. it is used for
further algebraic computations.
• It should have sampling stability i.e. if we take say 10 different samples
form the population it will result into almost same measures of central
values.
• It should not be unduly effected by the presence of extreme values

13. Absolute Vs. Relative Measures of Variations
• Absolute Measures of dispersion: Absolute measures of variations
are expressed in the same statistical units in which the original data
are given like rupees, kilogram, etc. These values are helpful in
comparing the variations in two or more distributions which have
almost same average value.
• Relative Measures of dispersion: Relative measures of dispersion are
useful in comparing two sets of data which have different units of
measurements. These are expressed as the percentage or the
coefficient of the absolute measure of dispersion.
Relative measures of variations is the ratio of measure of absolute
variation to an Average. It is also called coefficient of variation because
coefficient is a pure number that is independent of any unit of
measurements

14. Range
The range is the difference between highest value & the lowest value.
The symbol R is used for the range.
R = Highest Value - Lowest Value
Range is the absolute measure of dispersion. Relative measure of
range is
Coefficient of Range =
𝐿−𝑆
𝐿+𝑆

15. Range
Merits
• It is very easy to calculate and simple to understand.
• No special knowledge is needed while calculating range.
• It takes the least time for computation.
• It provides a broad picture of the data at a glance.
Demerits
• It is a crude measure because it is only based on two extreme values (highest and
lowest).
• It cannot be calculated in the case of open-ended series.
• Range is significantly affected by fluctuations of sampling, i.e. it varies widely
from sample to sample.
• Range cannot tells us anything about the characterisitics of the distribution.

16. Quartile Deviation
• It is known as semi-interquartile range, i.e., half of the
difference between the upper quartile and lower quartile.
• Quartile deviation can be calculated by:
QD = (Q3 – Q1)/2
Interquartile Range = Q3-Q1
Coefficient of Quartile deviation refers to the ratio of the
difference between Upper Quartile and Lower Quartile of a
distribution to their sum.
Coefficient of QD =
𝑸𝟑−𝑸𝟏
𝑸𝟑+𝑸𝟏

17. Quartile Deviation
Merits
• It is also quite easy to calculate and simple to understand.
• It can be used even in case of open-end distribution.
• It is less affected by extreme values so, it a superior to ‘Range’.
• It is more useful when the dispersion of the middle 50% is to be computed.
Demerits
• It is not based on all the observations.
• It is not capable of further algebraic treatment or statistical analysis.
• It is affected considerably by fluctuations of sampling.
• It is not regarded as a very reliable measure of dispersion because it
ignores 50% observations.

18. Average Deviation
• Mean deviation is the arithmetic mean (average) of deviations ⎜D⎜of
observations from a central value (mean or median).
A.D. =
𝛴 𝑥−𝑥
𝑛
or
𝛴 𝑥−𝑀𝑒𝑑𝑖𝑎𝑛
𝑛
• Coefficient of Mean Deviation from Mean =
𝐴𝐷
𝑥
• Coefficient of Mean Deviation from Median =
𝐴𝐷
𝑀𝑒𝑑.

19. Average Deviation
Merits
• It is based on all the observations of the series and not only on the limits like
Range and QD.
• It is simple to calculate and easy to understand.
• It is not much affected by extreme values.
• For calculating mean deviation, deviations can be taken from any average.
Demerits
• Ignoring + and – signs is bad from the mathematical viewpoint.
• It is not capable of further mathematical treatment.
• It is difficult to compute when the mean or median is in fraction.
• It may not be possible to use this method in case of open ended series.

20. Standard Deviation
• Standard deviation is the square root of the means of square
deviations from the arithmetic mean. It is also known as root mean
square deviation. This is given by Karl Pearson. It is denoted by “σ”
σ =
𝛴 𝑥−𝑥 2
ℕ
Variance = σ2

21. Coefficient of Variation
• It is used to compare two data with respect to stability (or uniformity
or consistency or homogeneity).
• It indicates the relationship between the standard deviation and the
arithmetic mean expressed in terms of percentage.
CV =
𝜎
𝑥
× 100
If coefficient is low distribution is more consistent, homogeneous and
uniform.

22. Standard Deviation
Merits
• It is the most popular measure of dlspersion in a distribution.
• It is a good measure of dispersion since all the values are used in its
computation.
• It is very important and useful in the testing of hypothesis..
• It is most useful mathematically, especially for further statistical analysis.
• It has great practical utility in sampling and statistical inference.
Demerits
• As compared to other measures of dispersion it is difficult to compute.
• It gives greater weightage to extreme values.eg two deviations of a series
are 2 and 10 then ratio is 1:5 but when we take squares of this deviations
it is 4 and 100 with ratio 1:25.

23. Measures of Shape
• These are the tools used for describing the shape of the distribution
of the data.
They are :
• Skewness
• Kurtosis

24. Skewness
• Skewness refers to lack of symmetry.
• When the distribution is not symmetrical (asymmetrical) it is known
as skewed distribution
• The measures of skewness indicate the difference between the
manner in which the observations are distributed in a particular
distribution compared with symmetrical distribution.

25. Concept of Skewness
A distribution is said to be skewed-when the mean, median and mode fall at
different position in the distribution and the balance (or center of gravity) is
shifted to one side or the other i.e. to the left or to the right.
Therefore, the concept of skewness helps us to understand the
relationship between three measures-
• Mean.
• Median.
• Mode.

26. Symmetrical Distribution
• A frequency distribution is said to be symmetrical if the frequencies
are equally distributed on both the sides of central value.
• A symmetrical distribution may be either bell – shaped or U shaped.
• In symmetrical distribution, the values of mean, median and mode are
equal i.e. Mean=Median=Mode

27. Skewed Distribution
• A frequency distribution is said to be skewed if the frequencies are not
equally distributed on both the sides of the central value.
• A skewed distribution maybe-
• Positively Skewed
• Negatively Skewed

28. Skewed Distribution
• Negatively Skewed
• In this, the distribution is skewed
to the left (negative)
• Here, Mode exceeds Mean and
Median.
• Positively Skewed
• In this, the distribution is skewed
to the right (positive)
• Here, Mean exceeds Mode and
Median.
Mean<Median<Mode Mode<Median<Mean

29. Type of Skewness
Positively skewed Negatively Skewed

30. Graphical Measures of Skewness
• Measures of skewness help us to know to what degree and in which direction (positive or negative)
the frequency distribution has a departure from symmetry.
• Positive or negative skewness can be detected graphically (as below) depending on whether the
right tail or the left tail is longer but, we don’t get idea of the magnitude
• Hence some statistical measures are required to find the magnitude of lack of symmetry
Mean=Median=Mode Mean<Median<Mode
Mean> Median> Mode
Symmetrical Skewed to the Left Skewed to the Right

31. Statistical Measures of Skewness
Absolute Measures of Skewness
Following are the absolute
measures of skewness:
• Skewness (Sk) = Mean – Median
• Skewness (Sk) = Mean – Mode
• Skewness (Sk) = (Q3 - Q2) - (Q2 -
Q1)
Relative Measures of Skewness
There are four measures of skewness:
• β and γ Coefficient of skewness
• Karl Pearson's Coefficient of skewness
• Bowley’s Coefficient of skewness
• Kelly’s Coefficient of skewness

32. β and γ Coefficient of Skewness

33. Karl Pearson's Coefficient of Skewness……01
• This method is most frequently used for measuring skewness. The formula
for measuring coefficient of skewness is given by
Where,
SKP = Karl Pearson's Coefficient of skewness,
σ = standard deviation.
SKP = Mean – Mode
σ
Normally, this coefficient of skewness lies between -3 to +3.

34. In case the mode is indeterminate, the coefficient of skewness is:
Now this formula is equal to
The value of coefficient of skewness is zero, when the distribution is symmetrical.
The value of coefficient of skewness is positive, when the distribution is positively skewed.
The value of coefficient of skewness is negative, when the distribution is negatively skewed.
SKP =
Mean – (3 Median - 2 Mean)
σ
SKP =
3(Mean - Median)
σ
Karl Pearson's Coefficient of Skewness…..02

35. Bowley’s Coefficient of Skewness……01
Bowley developed a measure of skewness, which is based on quartile values.
The formula for measuring skewness is:
Where,
SKB = Bowley’s Coefficient of skewness,
Q1 = Quartile first Q2 = Quartile second
Q3 = Quartile Third
SKB =
(Q3 – Q2) – (Q2 – Q1)
(Q3 – Q1)

36. Bowley’s Coefficient of Skewness…..02
The above formula can be converted to-
The value of coefficientof skewnessis zero, if it is a symmetrical distribution.
If the value is greater than zero, it is positively skewed distribution.
And if the value is less than zero, it is negatively skewed distribution.
SKB = Q3 + Q1 – 2Median
(Q3 – Q1)

37. Kelly’s Coefficient of Skewness…..01
Kelly developed another measure of skewness, which is based on percentiles and
deciles.
The formula for measuring skewness is based on percentile as follows:
Where,
SKK = Kelly’s Coefficient of skewness,
P90
P50
P10
= Percentile Ninety.
= Percentile Fifty.
= Percentile Ten.
SKk =
P10
P90 – 2P50 +
P90 – P10

38. Kelly’s Coefficient of Skewness…..02
This formula for measuring skewness is based on percentile are as follows:
Where,
SKK = Kelly’s Coefficient of skewness,
D9 = Deciles Nine.
D5 = Deciles Five.
D1 = Deciles one.
SKk = D9 – 2D5 + D1
D9 – D1

39. Moments:
•In Statistics, moments is used to indicate peculiarities of a frequency
distribution.
•The utility of moments lies in the sense that they indicate different
aspects of a given distribution.
•Thus, by using moments, we can measure the central tendency of a
series, dispersion or variability, skewness and the peakedness of the
curve.
•The moments about the actual arithmetic mean are denoted by μ.
•The first four moments about mean or central moments are following:-

40. Moments:
Moments around Mean Moments around any Arbitrary No

41. Conversion formula for Moments
(Mean)
(Variance)
(Skewness)
(Kurtosis)
1st moment:
2nd moment:
3rd moment:
4th moment:

42. Two important constants calculated from μ2, μ3 and μ4 are:-
β1 (read as beta one) β2 (read as beta two)

43. Kurtosis
•Kurtosis is another measure of the shape of a frequency curve. It is a Greek word, which
means bulginess.
•While skewness signifies the extent of asymmetry, kurtosis measures the degree of
peakedness of a frequency distribution.
•Karl Pearson classified curves into three types on the basis of the shape of their peaks.
These are:-
•Leptokurtic
•Mesokurtic
•Platykurtic

44. Kurtosis
• When the peak of a curve becomes
relatively high then that curve is
called Leptokurtic.
• When the curve is flat-topped,
then it is called Platykurtic.
• Since normal curve is neither very
peaked nor very flat topped, so it
is taken as a basis for comparison.
• This normal curve is called
Mesokurtic.

45. Measure of Kurtosis
• There are two measure of Kurtosis:
• Karl Pearson’s Measures of Kurtosis
• Kelly’s Measure of Kurtosis

46. Karl Pearson’s Measures of Kurtosis
Formula Result:

47. Kelly’s Measure of Kurtosis
Formula Result:

48. Measure of Kurtosis

49. Differences Between Skewness and Kurtosis
1- The characteristic of a frequency distribution that ascertains its symmetry
about the mean is called skewness. On the other hand, Kurtosis means the
relative pointedness of the standard bell curve, defined by the frequency
distribution.
2- Skewness is a measure of the degree of lopsidedness in the frequency
distribution. Conversely, kurtosis is a measure of degree of tailedness in the
frequency distribution.
3- Skewness is an indicator of lack of symmetry, i.e. both left and right sides of
the curve are unequal, with respect to the central point. As against this, kurtosis
is a measure of data, that is either peaked or flat, with respect to the probability
distribution.
4- Skewness shows how much and in which direction, the values deviate from
the mean? In contrast, kurtosis explain how tall and sharp the central peak is.