This document defines and explains various measures of central tendency, dispersion, and distribution used in descriptive statistics. It discusses modes, medians, means, percentiles, quartiles, range, interquartile range, standard deviation, z-scores, and other key statistical concepts. These metrics are used to summarize and describe the central position and variability of data in distributions.

1. Descriptive Statistics
Measure of Central Tendency
Measure of Dispersion
Measure of Distribution

2. Measure of Central Tendency:
Ungrouped Data
 Provides information about the center or
middle part of a group of numbers.
 Includes mean, median, mode, quartiles,
percentiles etc.

3. Mode
 Most frequently occurring value in a set of data.
 Less popular than mean and median
 Use to find out the value with highest demand in business.
How to determine the mode in a data set
 Order the values from minimum to maximum and locate the value which occurs the most.
 3,4,5,5,6,6,6,6,6,7,7,7,8,8,9,9,9,10,10,11,12 Mode=6
 3,4,5,5,5,5,6,6,6,6,8,8,9,9,9, 10, 11, 11, 12 Mode = 5 and 6 (Bimodal)
 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18 Mode = none

4. Median
 Middle value in an ordered array of numbers.
 For an array with an odd number of terms, the median is the
middle number
 For an array with an even number of terms, the median is an
average of the two middle numbers
 Steps to determine the median:
 Step1: Arrange the observation in an ascending/descending order
 Step2: For an odd number of terms, find the middle term
 Step3: For an even number of terms, the average of middle two
terms

5. Median Example: Calculate the median of the following example:
34 72 39 55 24 26 75 23 35 51 82 66 69 85 56 70 76 89 26 41
Step1:Arrange inascendingorder
2324262634353941515556666970727576828589
Step2:Themiddlevaluesare55and56
Step3:Medianis(55+56)/2= 55.5
Median is not affected by theextreme values
Median is not reflecting theinformation about all the numbers.

6. Mean
Arithmetic mean is the
average of a group of
members.
Computed by dividing
the sum of the numbers
by the total numbers.

7. Meanisaffectedbyallthevaluesinthedataset.
Meanisbadlyaffectedbytheoutliersortheextremevaluesinthe
dataset.

8. Percentiles
 99 values (dividers) which divide the
data into 100 equal parts.
 nth percentile means that n % of the
data is below than value. For example
87th percentile means 87% of the values
are lower than this number.
Percentiles are widely used in tests
such as CAT, JEE, GRE etc. The results of
these exams are reported in percentile
form along with raw scores.

9. Quartiles
Are measure of central tendency that divide a
group of data into four equal parts.
These quartiles are denoted as Q1, Q2 and Q3.
Q1 is 25th percentile, Q2 is 50th percentile and Q3
is 75th percentile.


10. Measure of Variability-
Ungrouped data
Describe the dispersion or
spread of the data set.
Provides significant information
along with measure of central
tendency.

11. Range and Interquartile range
Range is the difference between the largest value of a data set and the smallest value of the
data set.
Easy to compute
Not considered as a good measure as it considers the extreme values of the dataset.
Interquartile range is the difference between first quartile and third quartile of a dataset
i.e. Interquartile Range = Q3 - Q1
It indicates the range of 50% of the dataset.

12. Mean Absolute deviation
 Average of the absolute values of the deviations around the mean for a set of
numbers

13. Variation
 Average of the squared deviations from the mean for a set of numbers

14. Standard Deviation
 Square root of the variance

15. Chebyshev’s Theorem:
Helps in estimating the approximate percentage of values that lie within
a given number of standard deviation from the mean of a set of data if
the data is normally distributed.

16. Chebyshev’s theorem
statesthatatleast1–1/k2valueswillfallwithin+standarddeviationofthe
meanregardlessoftheshapeofthedistribution
Assuming the average weight of 56 kg and standard deviation of 10 kg.
Estimate that how much proportion of the population lie within the
range of mean + 2 * SD
a. If the distribution is normal.
b. If the distribution is not normal.

17. Z-Scores
is a numerical measurement used in statistics of a value's relationship to
the mean (average) of a group of values, measured in terms of standard
deviation from the mean.
If a Z-score is 0, it indicates that the data point's score is identical to the
mean score.
A Z-score of 1 would indicate a value that is one standard deviation
from the mean.
Z-scores may be positive or negative, with a positive value indicating the
score is above the mean and a negative score indicating it is below the
mean.

19. Coefficient of Variation
Is the ratio of standard deviation to the mean expressed in
percentage
𝐶𝑉 =
𝜎
𝜇
(100)
CV is a relative comparison of a SD to the mean.

20. Skewness

21. Kurtosis

22. Mean – Grouped data

23. Median Grouped Data

24. Box Plot Diagram

25. Thankyou