A graph is a pictorial representation of numerical data. Generally, the data used in statistics or other fields is vast and it is difficult to read all the data items and interpret the results. Graphical representation is very helpful in making comparisons, interpreting various results and even forecasting many possibilities. A graph is diagram which shows relationship between two variables. These variables are usually defined on X and Y axis. One variable is independent and is generally represented on X-axis, while the other variable is dependent and is generally represented on Y-axis.

## Types of Graphs and Charts

Graph and chart are ambiguous words. They make a difficult data easy to understand. One can interpret the results by just having a look at graph or chart. There are different types of graphs and charts. Few of them are listed below:
• Line graph
• Pie chart
• Bar graph or bar diagram
• Scatter plot or scatter graph
• Stem and leaf plot or stem plot
• Histogram
• Frequency polygon
• Frequency curve
• Cumulative frequency curve.

## Line Graph

A line graph is an important tool for understanding and comparing two variables. One variable is represented at X axis and other at Y axis. The points given are plotted and are joined together by straight lines. Following diagram is a line graph showing annual sales of a particular organization:

## Pie Chart

Pie chart is named so, due to its similarity with pie and its slices. Pie chart is a circular chart which is divided into sectors. Each sector corresponds to different values of a data. Generally, in pie charts, the percentage of each number is calculated and then obtained percentages are plotted at pie chart. Percentage of a data item can be calculated by the following formula:
Pie charts can represent measure of each sector in degree also.
Following pie chart shows exit poll results in an election:

## Bar Graph

Bar graph or bar chart is a graph which consists of rectangular bars. Length of these bars correspond to the values given in the data. Two consecutive rectangular bars are at fixed distance. Bar graphs are very useful and most suited for the qualitative data and discrete data. Following diagram shows a bar graph of marks obtained by students:

## Scatter Plot

Scatter plot is also known as scatter graph. This is a kind of graph in which only points are plotted on XY plane. Ideally, a dot or a small circle or a very small square is placed at the location of the point. Scatter plot is very useful when very large data is given. Scatter plot enables a researcher to conclude the result from a vast data set by just looking at the graph. One can immediately conclude if the data is collectively going up or down or being stable. A straight line can also be drawn which best fits the data. This line segment is called line of best fit. Line of best fit represents the whole set of data. Not every time, line of best fit is drawn. It depends upon the requirement.

Following graph shows scatter plot between age and weight of children:

## Stem and Leaf Plot

Stem and leaf plot is also called stem plot. It is a way of representing numerical or quantitative data. In order to draw a stem and leaf plot, first the data must be arranged in ascending order. Leaf contains digit at ones place and stem contains all other remaining digits. Then, two columns should be constructed. Stems are listed in left column and leaves are in right one.

A stem and leaf plot displaying marks obtained by few students in maths exam is constructed below:
45, 80, 78, 82, 71, 62, 55, 42, 89, 65, 79, 45, 82, 61, 90, 83, 95, 55, 71, 83

Ascending Order:
42, 45, 45, 55, 55, 61, 62, 65, 71, 71, 78, 79, 80, 82, 82, 83, 83, 89, 90, 95

Stem and Leaf Plot

 Stem Leaf 4 2 5 5 5 5 5 6 1 2 5 7 1 1 8 9 8 0 2 2 3 3 9 9 0 5
Stem and leaf plots were more used in early 80s than now. Stem and leaf plot is useful for a data which is not too small as well as not too large to be represented as a table. This plot can be constructed by hands easily and is very easy to interpret.

## Histogram

Histogram is a graphical representation which uses rectangular bars to present the data. Histograms look like bar graphs. Unlike bar graphs, histograms do not have gaps between two consecutive bars. Histograms are most useful for representing grouped data. Width of rectangular bar on X-axis usually denotes class interval.

Following graph demonstrates a histogram:

When a histogram represents frequencies (at Y-axis) of a given data, it is also called a frequency histogram.

## Frequency Polygon

Frequency polygon is an extension of histogram. Frequency polygon is constructed by marking mid points of all the bars in histogram and then, joining them together by straight lines. Though, frequency polygon can be made without drawing histogram by just marking midpoints (of given class interval) on X-axis and then plotting points corresponding to them, join these points with straight lines. Sometimes, it is beneficial to draw histogram and frequency polygon together.

Following graph shows a frequency polygon drawn by the same data used above in histogram:

## Frequency Curve

A curve is a line that is smooth and continuous. In a histogram, when midpoints are joined together with straight lines, a frequency polygon is formed. On the other hand, when midpoints are joined together by free hand resulting in a smooth curve, a frequency curve is formed. It means that the only difference between frequency polygon and frequency curve is that unlike frequency polygon, frequency curve is smooth and without sharp edges.

A diagram showing frequency curve of the same data used before is given below:

## Cumulative Frequency

Cumulative frequency of a particular frequency (in a given frequency distribution) is calculated by adding all the frequencies above to that particular frequency. Following table shows how cumulative frequency of a given data is computed:
 Days Number of cars parked (f) Cumulative Frequencies (c.f.) Monday 10 10 Tuesday 12 10 + 12 = 22 Wednesday 9 22 + 9 = 31 Thursday 8 31 + 8 = 39 Friday 15 39 + 15 = 54 Saturday 20 54 + 20 = 74 Sunday 18 74 + 18 = 92 Total = 92

Cross Check:
Total of all frequencies must be equal to last cumulative frequency in the table.
Cumulative frequency curve is a curve which is drawn by plotting cumulative frequency points on a graph. Cumulative frequency curve is also known as Ogive.

Following graph shows the cumulative frequency curve of above table:

## Graph Problems

Few problems based on graphs are given below:

### Solved Examples

Question 1: A student's scores in various subjects are recorded and following pie chart is drawn.

If his total score is 450, then answer the following two questions:
1. How much marks he obtained in mathematics and physics?
2. In how many subjects he scored less than the average marks obtained?

Solution:
1. Total score = 450
Measure of mathematics = 80$^o$
Marks obtained in mathematics = $\frac{80}{360}$$* 450 = 100 Measure of physics = 45^o Marks obtained in mathematics = \frac{45}{360}$$ * 450$
= 56.25 = 56
2. Total number of subjects = 6
Total measure = 360$^o$
Measure of average score = $\frac{360}{6}$ = 60$^o$
Measure of physics, social studies and GK are less than 60$^o$.
Therefore, in physics, social studies and GK, he scored less than average marks.

Question 2: Construct a cumulative frequency table of the following data and draw a cumulative frequency curve or an Ogive:
1, 2, 1, 7, 4, 5, 9, 9, 5, 4, 1, 9, 8, 8, 9, 1, 7, 5, 1, 9
Solution:
 Score Frequency (f) Cumulative Frequency (c.f.) 1 5 5 2 1 5 + 1 = 6 4 2 6 + 2 = 8 5 3 8 + 3 = 11 7 2 11 + 2 = 13 8 2 13 + 2 = 15 9 5 15 + 5 = 20 Total = 20
Using above data, following frequency distribution curve is obtained: