2. Your 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.
What is meant by data?
Any bit of information that is expressed in a value or numerical number is
data. For example, the number of cars that pass through a bridge in a day is
also data. Data is basically a collection of information, measurement or
observations.
INTRODUCTION TO GRAPHS
3. TYPES OF DATA
There are 3 types of data :
Raw data :
Raw data which is an initial collection of information this is the information which has not yet been
organized. the very first step of data collection is that you will collect raw data. For example, we go
around and ask a group of five friends their favorite colour. The answers are Blue, Green, Blue, Red,
and Red. This collection of information is called raw data.
Discrete data :
Discrete data is the 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.
Continuous data :
Continuous data. Continuous data need not be in whole numbers, it can be in decimals Example are
the temperature in a city for a week, your percentage of marks for the last exam.
4. FREQUENCY
What is frequency?
The frequency of any value is the number of times that value appears in a data set. 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.
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 cms.
139, 145, 150, 145, 136, 150, 152, 144, 138,
5. This frequency table will help us make better sense of the data given. Also when the data set is too
big (say if we were dealing with 100 students) we use tally marks for counting. It makes the task more
organised and easy. Below is an example of how we use tally marks.
Using the same above example we can make the following graph:
6. Types of frequency distribution
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 distribution :
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.
Example :
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
8. Ungrouped frequency distribution :
Then, a cumulative frequency distribution is 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.
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).
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 (percents).
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.
9. cumulative relative frequency distribution :
The cumulative relative frequency distribution of a quantitative variable is a summary of frequency
proportion below a given level.
The relationship between cumulative frequency and relative cumulative frequency is:
Why are frequency distributions important?
Frequency distribution 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.
10. Probability
What is probability?
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 .
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.
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 1:6. That was its probability.
11. Some concepts 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
What is 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 1, 2, 3,
4, 5, or 6. It cannot be predicted in advance, making the
rolling of dice a random experiment.
All possible outcomes or results of an experiment make up its
sample space. So the sample space of the above example will
be, S = { 1,,2,3,4,5,6}. Since a dice once thrown can give you
only one of these six results.
12. When a particular event occurs, like for example the dice lands on a six, we can say an
event has occurred. So we can say every possible outcome of a random experiment is
an event
Let us now change our example. Say you are now tossing an ordinary coin. Every time you
toss it either it lands 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.
13. 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
14. Arithmetic mean
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. So let us now study the topic arithmetic mean in detail.
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 example:
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.
15. Formula for Arithmetic Mean
Mean =Sum of all observations
Number of observations
Median and mode
Median
The number of students in your classroom, the money of money your parents earns, the
temperature in your city are all important numbers. But how can you get the information of
the number of students in your school or the amount earned by the citizen of your entire
city? 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 term :
Median = the middle number
16. 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 .
