Data Representation: Definition, Types and Examples

Data is the new engine oil therefore representing it effectively is an important part of making complex decisions. It helps in understanding complex decisions. The way data is represented can be helpful in proving different insights.

Representation of Data

Any bit of information that is expressed as a numerical number is data. For example, the marks you obtained in your Math exam are data. Data is a collection of information, measurements or observations.

The data once collected must be arranged or organized in a way so that inferences or conclusions can be made out from it.

The following are the ways to represent data

1. Ungrouped distribution
2. Ungrouped frequency distribution
3. Grouped frequency distribution

The frequency of any value is the number of times that value appears in a data set.

Ungrouped distribution

Consider the marks obtained (out of $100$ marks) by $30$ students of Class $XI$ of a school:

$\begin{equation}
\begin{array}{llllllllll}
10 & 20 & 36 & 92 & 95 & 40 & 50 & 56 & 60 & 70 \\
92 & 88 & 80 & 70 & 72 & 70 & 36 & 40 & 36 & 40 \\
92 & 40 & 50 & 50 & 56 & 60 & 70 & 60 & 60 & 88
\end{array}
\end{equation}$

This representation is called Ungrouped distribution, as all the values are simply mentioned and separated by comma.

Ungrouped Frequency Distribution

Observe that, $4$ students got $70$ marks. So the frequency of $70$ marks is $4$.

To make the data more easily understandable, we create a table,

\begin{array}{|c|c|}\hline \mathbf { Marks } & {\mathbf { Number\;of \;students }} \\ \hline 10 & {1} \\ 20 & {1} \\ {36} & {3} \\ {40} & {4} \\ {50} & {3} \\ {56} & {2} \\ {60} & {4} \\ {70} & {4} \\ {72} & {1} \\ {80} & {1} \\ {98} & {2} \\ {92} & {3} \\ {95} & {1} \\ \hline{\mathbf { Total }} & \mathbf{30} \\ \hline\end{array}

The above table is called a Ungrouped Frequency Distribution.

Grouped Frequency Distribution

We can show data as ranges of marks and the number of students that obtained marks in that range.

So we can represent this data as

\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Class interval } & {10-25} & {25-40} & {40-55} & {55-70} & {70-85} & {85-100} \\ \hline \text { Number of students } & {2} & {3} & {7} & {6} & {6} & {6} \\ \hline\end{array}

Here we have taken groups (range) of marks. This is why it is called Grouped distribution.

Also, the difference in marks in each interval is $15 (25-10 = 15, 70-55=15,...).$ This number is called the width of the class interval. Here width is 15, but we can take any width as per our convenience.

The above table is called a Grouped frequency distribution.

Relative frequency distribution

Relative frequency distribution is the distribution divided by the total number of observations.

$f_r-$ Relative frequency of a data or class interval
$f-$ frequency of a data or class interval
$N-$ sum of frequencies

$
f_r=\frac{f}{N}
$

Cumulative frequency distribution

The cumulative frequency of a value is its frequency plus the frequencies of all smaller values.

Example:

$ \begin{equation}
\begin{array}{ccc}
x & f & c . f \\
0 & 2 & 2 \\
1 & 4 & 6 \\
2 & 4 & 10 \\
3 & \underline{6} & 16 \\
& 16 &
\end{array}
\end{equation}$

The cumulative relative frequency of any value is similarly defined as the relative frequency of the value plus the relative frequencies of all smaller values.

Example:

$\begin{equation}
\begin{array}{ccc}
x & 1 / N=f_r & \text { c.r.f } \\
0 & 0.08 & 0.08 \\
1 & 0.16 & 0.24 \\
2 & 0.16 & 0.40 \\
3 & 0.24 & 0.64
\end{array}
\end{equation}$

Importance of Data Representations:

1. Proper data representations make data easier and clearer to understand.

2. Effectively represented data can be communicated broadly to a larger audience.

3. Decision-making relies on data representation properly.

Solved Examples Based on Representations of Data:

Example 1. What is the frequency of class interval $5$ - $10$ of the raw data - $ 3,2,1,5,6,9,0,10,7,8,10,5,6,7,3 $
1) $7$
2) $8$
3) $10$
4) $6$

Solution

As we learned

Discrete frequency distribution -

A representation of data in which each outcome is paired with its frequency is called a frequency distribution.

Example:

$\begin{equation}
\begin{array}{lllllll}
x: & 0 & 1 & 2 & 3 & 4 & 5 \\
\mathrm{f:} & 2 & 4 & 4 & 6 & 4 & 5
\end{array}
\end{equation}$

Class interval $5$ - $10$ has $\begin{equation}
5,6,9,7,8,6,7
\end{equation}$

We don't include $10$.

Hence, the answer is option 1.

Example 2: If the frequencies of the first four numbers out of $1, 2, 4, 6, 8$ are $2, 3, 3 , 2$ respectively, then the frequency of $8$ if their AM is $5$, is

1) $4$

2) $5$

3) $6$

4) none of these

Solution

As we learned

Discrete frequency distribution -

A representation of data in which each outcome is paired with its frequency is called a frequency distribution.

Here mean A=$5$

Let the frequency of $8$ be $x.$ Then by the formula

$\begin{equation}
\begin{aligned}
&\begin{aligned}
& A=\frac{\sum x f}{\sum f} \\
& 5=\frac{1.2+2.3+4.3+6.2+8 . x}{2+3+3+2+x}=\frac{32+8 x}{10+x}
\end{aligned}\\
&\text { of } 18=3 x ; \ x=6 \text {. }
\end{aligned}
\end{equation}$

Hence, the answer is an option 3.

Example 3: What is the interval width of data largest value = $20$, Last Value = $3$ number of intervals to splits= $3$?

1) $5$

2) $6$

3) $7$

4) does not exist

Solution

As we learned

Group frequency distribution -

The data is grouped into intervals: ${30-40, 40-50 ......}$ It reduces the cumbersome task of representing every value in the distribution as a unit as in discrete frequency distribution.

- wherein

Interval Width: $\frac{x_i-x_s+1}{n}$

Where $x_i$ is the largest value, $x_s$ is the smallest value and $n$ is a number of intervals to split data.

Interval width = $\frac{20-3+1}{3}=\frac{18}{3}=6$

Hence, the answer is the option (2).

Example 4: What is the relative frequency distribution for the table 5 - 10 ?

Class fi
$1-5$ $6$
$5-10$ $14$
$10-15$ $3$
$15-20$ $27$

Solution

As we learned

Relative frequency distribution -

Relative frequency distribution is the distribution divided by the total number of observations.

Since relative frequency $y=\frac{14}{6+14+3+27} \times 100\%=\frac{14}{50} \times 100\% = 28\%$

Example 5: Which class interval has maximum relative frequency?

$\begin{equation}
\begin{array}{ll}
\text { class } & \text { frequency } \\
0-2 & 3 \\
2-4 & 2 \\
4-6 & 5 \\
6-8 & 8 \\
8-10 & 2 \\
10-12 & 1
\end{array}
\end{equation}$

1) $0-4$
2) $2-6$
3) $4-8$
4) $6-10$

Solution

As we learned

Relative frequency distribution -
$
f_r=\frac{f}{N}
$
wherein
$f_r-$ Relative frequency of a data or class interval
$f-$ frequency of a data or class interval
$N-$ sum of frequencies
$
f r=\frac{f}{N}
$
$4-8$ has $13 ; 6-10$ has $10$ frequencies.
Thus $4$ - $8$ is the answer.

Hence, the answer is option 3.

Summary

Data is a collection of information, measurements or observations. The data once collected must be arranged or organized in a way so that inferences or conclusions can be made out from it. There are 3 representations of data, namely, Ungrouped distribution, Ungrouped frequency distribution, Grouped frequency distribution.