This document contains a summary of key statistical concepts and methods for analyzing grouped data. It includes 10 topics: (1) names of group members, (2) acknowledgements, (3) an introduction to statistics, (4) grouped data, (5) mean of grouped data, (6) mode of grouped data, (7) median of grouped data, (8) graphical representation of cumulative distribution, (9) conclusion, and (10) bibliography. The document provides examples and explanations of statistical techniques for summarizing and visualizing grouped data, including calculating the mean, mode, and median from frequency tables and constructing cumulative frequency distributions in ogive graphs.

1. SUNNY TRIPATHI
RANDHIR KUMAR
ARIHANT PANDEY
 VIKAS YADAV
SAHIL ALAM

2. S.NO TOPIC SLIDE NO
1. NAME OF GROUP MEMBER 1
2. ACKNOWLEDGEMENT 3
3. INTRODUCTION TO STATISTICS 4-6
4. GROUPED DATA 7-9
5. MEAN GROUPED OF DATA 10-11
6. MODE GROUPED OF DATA 12-18
7. MEDIAN GROEPED OF DATA 19-20
8. GRAPHICAL REPERSENTATION OF CUMULATIVE
DISTRIBUTION
21-25
9. CONCLUSION 26
10. BIBLIOGRAPHY 27

3. ACKNOWLEDGEMENT
 One cannot succeed alone no matter how great one’s abilities are,
without the cooperation of others. This project, too, is a result of efforts
of many. I would like to thank all those who helped me in making this
project a success.
 I would like to express my deep sense of gratitude to my Maths Teacher,
Mrs. H.David who was taking keen interest in our lab activities and
discussed various methods which could be employed towards this
effect, and I really appreciate and acknowledge her pain taking efforts in
this endeavour.

4. INTRODUCTION TO STATISTICS
 Statistics is the study of the collection, analysis, interpretation, presentation, and
organization of data. In applying statistics to, e.g., a scientific, industrial, or social
problem, it is conventional to begin with a statistical population or a statistical
model process to be studied. Populations can be diverse topics such as "all people
living in a country" or "every atom composing a crystal". Statistics deals with all
aspects of data including the planning of data collection in terms of the design of
surveys and experiments
 Some popular definitions are:
 Merriam-Webster dictionary defines statistics as "classified facts representing the
conditions of a people in a state – especially the facts that can be stated in
numbers or any other tabular or classified arrangement".
 Statistician Sir Arthur Lyon Bowley defines statistics as "Numerical statements of
facts in any department of inquiry placed in relation to each other".

5.  When census data cannot be collected, statisticians collect data by developing specific
experiment designs and survey samples. Representative sampling assures that inferences
and conclusions can safely extend from the sample to the population as a whole. An
experimental study involves taking measurements of the system under study,
manipulating the system, and then taking additional measurements using the same
procedure to determine if the manipulation has modified the values of the
measurements. In contrast, an observational study does not involve experimental
manipulation.
 Two main statistical methodologies are used in data analysis: descriptive statistics, which
summarizes data from a sample using indexes such as the mean or standard deviation,
and inferential statistics, which draws conclusions from data that are subject to random
variation (e.g., observational errors, sampling variation).[4] Descriptive statistics are most
often concerned with two sets of properties of a distribution (sample or population):
central tendency (or location) seeks to characterize the distribution's central or typical
value, while dispersion (or variability) characterizes the extent to which members of the
distribution depart from its center and each other. Inferences on mathematical statistics
are made under the framework of probability theory, which deals with the analysis of
random phenomena.

6.  A standard statistical procedure involves the test of the relationship between two statistical data sets, or a
data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the statistical
relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis
of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical
tests that quantify the sense in which the null can be proven false, given the data that are used in the test.
Working from a null hypothesis, two basic forms of error are recognized: Type I errors (null hypothesis is
falsely rejected giving a "false positive") and Type II errors (null hypothesis fails to be rejected and an actual
difference between populations is missed giving a "false negative").[5] Multiple problems have come to be
associated with this framework: ranging from obtaining a sufficient sample size to specifying an adequate
null hypothesis.[citation needed]
 Measurement processes that generate statistical data are also subject to error. Many of these errors are
classified as random (noise) or systematic (bias), but other types of errors (e.g., blunder, such as when an
analyst reports incorrect units) can also be important. The presence of missing data and/or censoring may
result in biased estimates and specific techniques have been developed to address these problems.
 Statistics can be said to have begun in ancient civilization, going back at least to the 5th century BC, but it
was not until the 18th century that it started to draw more heavily from calculus and probability theory.
Statistics continues to be an area of active research, for example on the problem of how to analyze Big data.

7. Grouped Data
 Grouped data is a statistical term used in data analysis. A raw dataset can be
organized by constructing a table showing the frequency distribution of the
variable (whose values are given in the raw dataset). Such a frequency table is
often referred to as grouped data.
 The idea of grouped data can be illustrated by considering the following raw
dataset:
Table 1: Time taken (in seconds) by a group of students to
answer a simple math question.
20 25 24 33 13
26 8 19 31 11
16 21 17 11 34
14 15 21 18 17

8.  The above data can be organised into a frequency distribution (or a grouped
data) in several ways. One method is to use intervals as a basis.
The smallest value in the above data is 8 and the largest is 34. The interval
from 8 to 34 is broken up into smaller subintervals (called class intervals). For each
class interval, the amount of data items falling in this interval is counted. This
number is called the frequency of that class interval. The results are tabulated as a
frequency table as follows:
Table 2: Frequency distribution of the time taken (in seconds) by the group of
students to
answer a simple math question
Time taken (in seconds) Frequency
5 ≤ t < 10 1
10 ≤ t < 15 4
15 ≤ t < 20 6
20 ≤ t < 25 4
25 ≤ t < 30 2
30 ≤ t < 35 3

9.  Another method of grouping the data is to use some qualitative characteristics
instead of numerical intervals. For example, suppose in the above example,
there are three types of students: 1) Below normal, if the response time is 5 to
14 seconds, 2) normal if it is between 15 and 24 seconds, and 3) above normal if
it is 25 seconds or more, then the grouped data looks like:
Table 3: Frequency distribution of the three types of students
Frequency
Below normal 5
Normal 10
Above normal 5

10. Mean of grouped data
 An estimate, of the mean of the population from which the data are drawn can
be calculated from the grouped data as:
 In this formula, x refers to the midpoint of the class intervals, and f is the class
frequency. Note that the result of this will be different from the sample mean of
the ungrouped data. The mean for the grouped data in the above example, can
be calculated as follows:

11. Class Intervals Frequency ( f ) Midpoint ( x ) f x
5 and above, below 10 1 7.5 7.5
10 ≤ t < 15 4 12.5 50
15 ≤ t < 20 6 17.5 105
20 ≤ t < 25 4 22.5 90
25 ≤ t < 30 2 27.5 55
30 ≤ t < 35 3 32.5 97.5
TOTAL 20 405
Thus, the mean of the grouped data is

12. Mode of Grouped Data
 The mode is the "most popular" number in a given data. The expression
comes from the French "a la mode" meaning "fashionable".
In vital statistics we study the numerical records of marriages, births, sickness,
deaths, etc. With the help of these records, the health and the growth of a
community may be studied for vital statistics which constitutes a large part of the
subject known as demography.
In demography we are concerned with the growth of human population.

13.  Growth of population is divided into three main components.
(i) Fertility (positive component)
(ii) Mortality (negative component)
(iii) Migration (positive and negative component)
In these components we will be able to get the maximum value which gives us the
modal value.
 The mode or modal value is that value in a series of observations which occur with
the greatest frequency e.g. the mode of the series 3, 5, 8, 5, 4, 5, 9, 3 would be
5,since this value occurs most frequently than any of the other values.
The mode of a distribution is the value at the point around which the items tend to
be most heavily concentrated. It is regarded as the most typical value of distribution.
The following diagram shows the modal value:

14.  The value of the variable for which the curve reaches the maximum is called the
mode. It is the value around which the items tend to be greatly concentrated.
There are many situations in which arithmetic mean and median fail to reveal
the true characteristics of data.
Example : When we talk of the most common wage, Income, height,
size of the shoe or readymade garment we have to find out mode,
because in these cases Arithmetic mean and median cannot represent
the data.
For example if the data are
Size of
Shoes
5 6 7 8 9 10 11
No. of
Person
10 20 25 40 22 15 6

15.  Now in this case if we have to find out the average shoe size. Arithmetic mean
cannot represent it. This series can be represented by mode.
The modal size is 8, since more persons are wearing this size compared to
others.
 Calculation of Mode
For determining mode, count the number of times the various values repeat
themselves. The value that occurs the maximum number of times is the modal
value. The more the modal value appears relatively, the more valuable it is, as an
average to represent mode.
Roll No 1 2 3 4 5 6 7 8 9 10
Marks 10 27 24 12 27 27 20 18 15 28
Example
Calculate the mode from the following data

16. Marks No. of times it occur
10 1
12 1
15 1
18 1
20 1
24 1
27 3
28 1
Solution
Calculation of mode
Since the item 27 occurs the maximum number of
times, i.e. 3, the modal marks are 27.

18. Example
The following is the distribution of height of students of a certain class in a
certain city:
Height (in cm) 160-162 163-165 166-168 169-171 171-173
No. of students 15 118 142 127 18
Find the average height of maximum number of students.
Solution
If the intervals are not continuous, 0.5 should be subtracted from the lower limit
and 0.5 should be added to the upper limit. Then the distribution table is
Height (in cm) 159.5-162.5 162.5-165.5 165.5-168.5 168.5-171.5 171.5-173.5
No. of students 15 118 142 127 18

19. Median of Grouped Data
a) The median is the middle of a distribution: One half of the scores are
above the median and the other half below the median. The median is less
sensitive to extreme scores than the mean and this makes it a better
measure than the mean for highly skewed distributions. The mean,
median, and mode are equal in symmetric distributions.
b) If the values of xi in a raw data are arranged in ascending or descending
order, then the middle-most value in the arrangement is called the median.
c) Arrange the values in ascending or descending order of magnitude.
For computation of median of a ungrouped data, proceed as
follows:
1. Take the middle-most value of the arrangement as the median.
2. If the number of values (n) in the raw data is odd, then the median will
be the th value of the arrangement

20. 3. If the number of values (n) in the raw data is even, then the average
of two middle most values th and th will determine the median.
The arithmetic mean of these two values will give the exact
median.
Example
Find the median of the following values of a variate:
10, 2, 3, 2, 5, 7, 9, 11, 6
Solution
Arranging the values in the ascending order. We get:
2, 2, 3, 5, 6, 7, 9, 10, 11
Here number of observations
n = 9 (odd)
∴Median = size of the th term
∴Median = size of the th term
= size of the 5th term = 6

21. Graphical Representation of Cumulative
Frequency Distribution
There are different types of graphical representation of statistical data.
a) Bar graphs
b) Histogram
c) Frequency polygon
d) Cumulative frequency Distribution
Now we can learn in detail about Cumulative frequency distribution as you
have learned all other graphical representations
Cumulative frequency distribution
Cumulative frequency is obtained by adding the frequency of a class interval
and the frequencies of its preceding intervals upto that class interval.
This is explained by an example below.
Daily income (in Rs) 100-120 120-140 140-160 160-180 180-200
Number of workers 12 14 8 6 10

22. Daily Income (in Rs) Daily Income (in Rs)
(Upper Limit)
Number of
Workers
Cumulative Frequency
100 - 120 Less than 120 12 12
120 - 140 Less than 140 14 12 + 14 = 26
140 - 160 Less than 160 8 26 + 8 = 34
160 - 180 Less than 180 6 34 +6 = 40
180 – 200 Less than 200 10 40 + 10 =50
The above distribution is called ‘less than’ cumulative frequency
distribution.
To represent the data graphically,
1) Mark the upper limits of the class interval on the x – axis and the
corresponding cumulative frequencies on the y − axis choosing suitable
scale.
2) Plot the points with coordinates having abscissa as upper limits and
ordinates as the cumulative frequencies.
3) Join the points by a free hand smooth curve

23. 4) The curve we get is called ‘Cumulative frequency curve’ or ‘less than ogive’
This graphical representation of the frequency distribution is called Ogive.

24. Now we can see the ‘more than’ cumulative frequency distribution
Daily Income (in Rs) Daily Income (in Rs)
(Lower Limit)
Number
of
Workers
Cumulative Frequency
100 - 120 More than or equal to
100
12 50
120 - 140 More than or equal to
120
14 50 - 12 = 38
140 - 160 More than or equal to
140
8 38 – 14 = 24
160 - 180 More than or equal to
160
6 24 – 8 = 16
180 – 200 More than or equal to
180
10 16 – 6 = 10
To represent the data graphically,
1) Mark the lower limits of the class interval on the x – axis and the corresponding
cumulative frequencies on the y − axis choosing suitable scale.

25. 2) Plot the points with coordinates having abscissa as lower limits and
ordinates as the cumulative frequencies.
3) Join the points by a free hand smooth curve.
4) The curve we get is called ‘Cumulative frequency curve’ or ‘more than
ogive’.

26. CONCLUSION
Statistics is a study of knowledge which different methods of collection,
classification, presentation, analysis, and interpretation of data.
Statistical application is very important in the field of research.

27. BIBLIOGRAPHY
 We have taken help from the INTERNET and NCERT BOOK in making this
project
 URL: www.Wikipedia.org/statistics
 www.gradestack.com/class10/completecourse/Statistics
 BOOK NAME: Ncert Math Class 10