This document provides an introduction to data handling and various statistical concepts. It defines different types of data like raw data, discrete data and continuous data. It then discusses frequency and different types of frequency distributions like grouped, ungrouped, cumulative, relative and relative cumulative distributions. It also explains concepts related to probability, chance and the probability formula. Finally, it covers topics like arithmetic mean, median and mode and provides examples to illustrate these statistical concepts.

1. DATA HANDLING
SEA OF MATHS
BY: HIMAKSHI, DEV, ALANKRITA, DEVANSHEE

2. INTRODUCTION
Our life is surrounded by numbers. Marks
you scored, runs you made, your height, your
weight. All these numbers are nothing but
data. Here we will learn what is data and how
to organise and represent this data in forms
of graphs and charts. We will also come
across the concept of probability. Let us get
started.

3. DATA AND IT’S TYPES
First is raw data which is an initial collection of
information. This information has not yet been
organized.
second type of data is Discrete data which is recorded in
whole numbers, like the number of children in a school or
number of tigers in a zoo. It cannot be in decimals
or fractions.
the third type is Continuous data. Continuous data need not be in
whole numbers, it can be in decimals. Examples are the
temperature in a city for a week, your percentage of marks for the
last exam etc.
Any bit of information that is expressed in a value or
numerical number is data. For example, the marks you
scored.

4. FREQUENCY AND IT’S TYPES
The frequency of any value is the number of times that value
appears in a data set.
To understand it we will first take an example:-
We go around and ask a group of five friends their favourite
colour. The answers are Blue, Green, Blue, Red, and Red. So
from the above examples of colours, we can say two children
like the colour blue, so its frequency is two. So to make
meaning of the raw data, we must organize. And finding out the
frequency of the data values is how this organisation is done.

5. FREQUENCY DISTRIBUTION
Many times it is not easy or feasible to find the frequency of data
from a very large dataset. So to make sense of the data we make a
frequency table and graphs. Let us take the example of the heights
of ten students in centimetres they are 139, 145, 150, 145, 136, 150,
152, 144, 138, 138
This frequency table will help us make better sense of the data
given

6. TYPES OF FRQUENCY DISTRIBUTIONS
There 5 types of Frequency distributions which is :
 Grouped frequency distribution.
 Ungrouped frequency distribution.
 Cumulative frequency distribution.
 Relative frequency distribution.
 Relative cumulative frequency distribution.

7. GROUPED FREQUENCY:
These are the numbers of
newspapers sold at a local shop
over the last 10 days: 22, 20, 18, 23,
20, 25, 22, 20, 18, 20 Let us count
how many of each number there is:
Papers Sold Frequency
 18 2
 19 0
 20 4
 21 0
 22 2
 23 1
 24 0
 25 1
It is also possible to group the
values. Here they are grouped in 5s:
Papers Sold Frequency
 15-19 2
 20-24 7
 25-29 1
Grouped frequency is the
frequency where several numbers
are grouped together. Grouped
frequency distribution helps to
organize the data more clearly. It
is more useful when the scores
have multiple values.

8. Ungrouped frequency
Let the test scores of all 20 students be
as follows:
23, 26, 11, 18, 09, 21, 23, 30, 22, 11, 21,
20, 11, 13, 23, 11, 29, 25, 26, 26
Marks obtained in the test No. of
students (Frequency)
 09 1
 11 4
 13 1
 18 1
 20 1
 21 2
 22 1
 23 3
 25 1
 26 3
 29 1
 30 1
 Total 20
It is just the opposite of grouped
frequency.

9. Cumulative frequency
distribution
Here’s a simple example:
You get paid $250 for a
week of work. The second
week you get paid $300 and
the third week, $350. Your
cumulative amount for week
2 is $550 ($300 for week 2
and $250 for week 1). Your
cumulative amount for week
3 is $900 ($350 for week 3,
$300 for week 2 and $250
for week 1).
The sum of the class and all
classes below it in a
frequency distribution. All
that means is you’re adding
up a value and all of the
values that came before it.

10. Relative frequency
distribution
 This relative frequency
distribution table shows how
people’s heights are
distributed.
 Note that in the right column,
the frequencies (counts) have
been turned into relative
frequencies (percent's). How
you do this:
 Count the total number of
items. In this chart the total is
40.
 Divide the count (the
frequency) by the total
number. For example, 1/40 =
.025 or 3/40 = .075.
In this we don’t want to
know the counts. We want
to know the percentages. In
other words, what
percentage of people used
a particular form of
contraception?

11. cumulative relative
frequency distribution of
a quantitative variable
 The relationship between
cumulative frequency and
relative cumulative
frequency is:
Cumulative frequency =
𝑪𝒖𝒎𝒖𝒍𝒂𝒕𝒊𝒗𝒆 𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚
𝑺𝒂𝒎𝒑𝒍𝒆 𝒔𝒊𝒛𝒆
It is a summary of
frequency proportion below
a given level.

12. WHY ARE FREQUENCY
DISTRIBUTIONS IMPORTANT ?
It has great importance in statistics. Also, a
well-structured frequency distribution makes
possible a detailed analysis of the structure
of the population with respect to given
characteristics. Therefore, the groups into
which the population break down can be
determined.

13. Chance and probability
Let us explain both these
concepts with an example.
You have gathered your
friends to come and play a
friendly board game. It is your
turn to roll the dice. You really
need a six to win the whole
game. Is there any way
to guarantee that you will roll
a six? Of course, there isn’t.
What are the chances you will
roll a six? Well if you apply the
basic logic you will realize you
have a one in six chance of
rolling a six.
Do you ever leave anything to
‘chance’? Like perhaps leave out a
chapter from your revision
because it ‘probably’ won’t come
in an exam? These terms ‘chance’
and ‘probability‘ can actually be
expressed in mathematical terms.
Come let us take a closer look at
probability and the probability
formula.

14. AN EXAMPLE
You have gathered your friends to come and play a friendly
board game. It is your turn to roll the dice. You really need a six
to win the whole game. Is there any way to guarantee that you
will roll a six? Of course, there isn’t. What are the chances you
will roll a six?
Well if you apply the basic logic you will realize you have a one
in six chance of rolling a six.
Now based on the above example let us look at some concepts
of probability. Probability can simply be said to be the chance
of something happening, or not happening. So the chance of
an occurrence of a somewhat likely event is what we call
probability. In the example given above the chance of rolling a
six was one is six. That was its probability.

15. CONCEPT RELATED TO
PROBABILITY
First is Random experiment
Second is Sample space
Third is an Event
Fourth is Equally likely Events
Fifth is Occurrence of an Event

16. RANDOM EXPERIMENT
A process which results in some well-defined
outcome is known as an experiment. Here you
rolling the dice was the random experiment, since
the outcome was not sure. The outcome here is 1, 2,
3, 4, 5, or 6. It cannot be predicted in advance,
making the rolling of dice a random experiment.

17. SAMPLE SPACE
Let us now change our example. Say you are now tossing an
ordinary coin. Every time you toss it either land on heads or on
tails. Every time the coin gets tossed there is a 50% chance of
heads and 50% chance of tails. Both events are equally likely,
i.e. they have an equal chance of happening. This is what we
call Equally likely events.
A particular event will be said to occur if this event E is a part
of the Sample space S, and such an event E actually happens.
So in the above experiment, if you actually roll a six, the event
will have occurred. A particular event will be said to occur if
this event E is a part of the Sample space S, and such an event
E actually happens. So in the above experiment, if you actually
roll a six, the event will have occurred.

18. THE FORMULA
Now that we have seen the concepts related to probability, let
us see how it is actually calculated. To see what are the
chances that an event will occur is what probability is. Now it is
important to remember that we can only
calculate mathematical probability of a random experiment. The
equation of probability is as follows:
P = Number of desirable events ÷ Total number of outcomes
Using this formula let us calculate the probability of the above
example. Here the desirable event is that your dice lands on a
six, so there is only one desirable event. And the
total number of possible results, i.e. the sample space, is six.
So we can calculate the probability, using the probability
formula as, P = 1/6

19. Arithmetic mean
In general language arithmetic
mean is same as the average
of data. It is the
representative value of the
group of data. Suppose we are
given ‘ n ‘ number of data and
we need to compute the
arithmetic mean, all that we
need to do is just sum up all
the numbers and divide it by
the total numbers. Let us
understand this with an
Suppose the principal of your
school asks your class teacher
that how was the score of your
class this time? What do you
think is the teacher going to do?
Do you think that the teacher is
going to actually read out the
individual score of all the
students? NO!!! What the teacher
does is, the teacher will tell the
average score of the class
instead of saying the individual
score. So the principal gets an
idea regarding the performance
of the students.

20. FORMULA TO CALCULATE MEDIAN
There are two sisters, with different heights. The height of the
younger sister is 128 cm and height of the elder sister us
150cm. So what if you want to know the average height of the
two sisters? What if you are asked to find out the mean of the
heights? As their total height is divided into two equal parts,
So 139 cm is the average height of the sisters. Here 150 > 139
> 128. Also, the average value also lies in between the
minimum value and the maximum value.
Formula for Arithmetic Mean
Mean =Sum of all observations
Number of observations

21. Median and mode  The median number varies
according to the total
number being odd or even.
Initially let us assume the
number as the odd number.
Now if we have numbers like
12, 15, 21, 27, 35. So here we
can say that the midpoint
here is 21,
The number of students in your
classroom, the money of money your
parents earns, are all important numbers.
But how can you get the information of
the number of students in your school .
This is where median and mode comes is
useful. So let us now study median and
mode in detail. To define the median in
one sentence we can say that the median
gives us the midpoint of the data. What do
you mean by the midpoint? Suppose you
have ‘ n ‘ number of data, then arrange
these numbers in ascending or
descending order. Just pick the midpoint
from the particular series. The very first
thing to be done with raw data is to
arrange them in ascending or descending
order. In Layman’s terms:
Median = the middle number

22. THANK YOU