## Graphical Presentation of Data

Tabulation and grouping does make data simple to understand and analyze. However, just the numerical data is not attractive enough to present it to higher management, stakeholders and those not very familiar with the particular functional area.

Moreover, pictorial or graphical representation is catchy to appreciate, remember, and grasp quickly and easy to explain. It allows us to obtain the underlying information in one glance. “One picture is equal to a thousand words” as the proverb goes.

Table of Contents

Hence, diagrams, graphs and charts have assumed importance for decision-making to the managers. To communicate the information effectively to the higher management, you must present the data in pictorial format whenever feasible, and support it with the numerical data as a reference.

Remember, higher management may not have adequate time to analyze the numerical data. Similarly, always present the information to junior employees as diagrams, graphs and charts, because they may not have adequate knowledge and grasp of numerical analysis.

**Difference Between Diagrams and Graphs****Types of Diagrams****Bar Diagram****Histogram****Pie Diagram****Frequency Polygon****Ogives**

### Difference Between Diagrams and Graphs

A brief distinction between a diagram and a graph is given below in Table.

Diagram | Graph |
---|---|

1. Can be drawn on an ordinary paper | 1. Can be drawn on graph paper. |

2. Easy to grasp. | 2. Needs some effort to grasp |

3. Not capable of analytical treatment. | 3. Capable of analytical treatment. |

4. Can be used only for comparisons | 4. Can be used to represent a mathematical relation. |

5. Data are represented by bars, rectangles, pictures, etc. | 5. Data are represented by lines curves. |

A graphic presentation is used to represent two types of statistical data: (i) Time Series Data and (ii) Frequency Distribution.

### Types of Diagrams

There are a large number of diagrams that can be used for the presentation of data. The selection of a particular diagram depends upon the nature of the data, the objective of the presentation, and the ability and experience of the person doing this task.

For convenience, various diagrams can be grouped under the following categories:

#### One-dimensional Diagrams

One-dimensional diagrams are also known as bar diagrams. In the case of one-dimensional diagrams, the magnitude of the characteristics is shown by the length or height of the bar. The width of a bar is chosen arbitrarily so that the constructed diagram looks more elegant and attractive.

It also depends upon the number of bars to be accommodated in the diagrams. If large numbers of items are to be included in the diagram, lines may also be used instead of bars.

#### Two-dimensional Diagrams

In the case of a two-dimensional diagram, the value of an item is represented by an area. Such diagrams are also known as ‘surface’ or ‘area diagrams’. Popular forms of two-dimensional diagrams are:

- Rectangular Diagrams
- Square Diagrams
- Circular or Pie Diagrams.

#### Three-dimensional Diagrams

With the help of three-dimensional diagrams, the values of various items are represented by the volume of a cube, sphere, cylinder, etc. These diagrams are normally used when the variations in the magnitudes of observations are very large.

#### Pictograms and Cartograms

These are like frequency plots. The data points are plotted on the graph in the same manner. Then instead of joining the data points, pictures or objects of the height of the data points are used to depict the data.

In that case, the heights of the pictures or objects represent the frequency. These include Histograms and frequency polygons.

### Bar Diagram

Bar diagrams and Column diagrams are very common in representing business data. These are used to depict the frequencies of different categories of variables. In the case of bar diagrams the bars are horizontal with their lengths proportional to the frequencies.

On the other hand, in column diagrams, the frequencies are depicted by vertical columns having their length proportional to the frequencies. We can also have multiple bars or columns representing different categories of variables.

Further, data related to sub-categories in a category can be shown on the same bar or column by overlapping the bars or columns on top.

**Example**: Draw a multiple-bar diagram to present the following data. Also, draw a multiple-column diagram.

Year | Sales (‘000 ₹) | Gross Profit (‘000 ₹) | Net Profit (‘000 ₹) |
---|---|---|---|

1996 | 120 | 40 | 20 |

1997 | 135 | 45 | 30 |

1998 | 140 | 55 | 35 |

1999 | 150 | 60 | 40 |

**Solution**:

**Bar Diagram:** We take the year on the Y axis and the rupees in thousands on the X axis. Then we draw horizontal bars with lengths proportional to the values of variables ‘Sales’, ‘Gross Profits’, and ‘Net Profits’. The bar diagram for the above data is as follows:

**Column Diagram:** We take the year on the X axis and the rupees in thousands on the Y axis. Then we draw vertical columns with lengths proportional to the values of variables ‘Sales’, ‘Gross Profits’, and ‘Net Profits’. The column diagram for the above data is as follows:

**Scatter Diagram: A scatter** diagram is the most fundamental graph plotted to show the relationship between two variables. It is a simple way to represent bivariate distribution. Bivariate distribution is the distribution of two random variables. Two variables are plotted one against each of the X and Y axis.

Thus, every data pair of (xi , yj ) is represented by a point on the graph, x being abscissa and y being the ordinate of the point. From a scatter diagram we can find if there is any relationship between the x and y, and if yes, what type of relationship. A scatter diagram thus indicates the nature and strength of the correlation.

**Example**: Draw a scatter diagram for the following data of eight years between income (X) and expenditure (Y).

Income (X) (₹) | 100 | 110 | 113 | 120 | 125 | 130 | 130 | 140 |

Expenditure (Y) (₹) | 85 | 90 | 91 | 100 | 110 | 125 | 125 | 130 |

**Solution**:

**Line Diagram:** It is similar to the frequency polygon, where we plot one or more variables against one variable. One variable against which other variables are plotted is taken along the X-axis.

It is commonly used to depict the trends in anytime series data. We can show one or more variables like economic, market trends, financial results, etc. together so that these can be compared.

**Example**: Draw a line diagram to present the following data.

Year | Sales (‘000 ₹) | Gross Profit (‘000 ₹) | Net Profit (‘000 ₹) |
---|---|---|---|

1996 | 120 | 40 | 20 |

1997 | 135 | 45 | 30 |

1998 | 140 | 55 | 35 |

1999 | 150 | 60 | 40 |

**Solution:** We take the year on the X axis and the rupees in thousands on the Y axis. Then we plot the data points for the variables ‘Sales’, ‘Gross Profits’, and ‘Net Profits’. These data points are then joined by straight lines to draw the line diagram. The line diagram for the above data is as follows:

### Histogram

Besides the frequency polygon, the histogram is one of the most popular and widely used graphical representations. It uses vertical bars whose height represents the frequency. In the histogram, the vertical bars touch the neighboring bars sharing one edge.

Hence, if the data is of inclusive classes, it needs to be converter to exclusive classes so that the class boundaries overlap. Sometimes, we also use histograms superimposed with frequency polygons. This helps interpolation of data, at the same time retaining the attractive representation of the histogram.

**Example**: In a city, the income tax department had the data as follows for the number of taxpayers along with the range of income tax they paid for a particular year. Represent the data graphically with the help of a histogram.

Tax paid in ₹ ‘000 | 20-24 | 25-29 | 30-34 | 35-39 | 40-44 | 45-49 | Total |

Number of Tax Payers | 45 | 130 | 200 | 65 | 45 | 15 | 500 |

Solution: For plotting the data we will first convert the data as exclusive classes. This is done by increasing the upper limits and decreasing the lower limits by an amount equal to half of the difference between the upper limit of any class and the lower limit of the subsequent class.

This makes the class boundary to join. The class boundaries of tax-paid classes are plotted on the X axis and the number of taxpayers on the Y axis. Then vertical bars are drawn of widths equal to classes and heights equal to the frequencies of corresponding classes. This is depicted as follows.

Tax paid in ₹ ‘000 | 19.5-24.5 | 24.5-29.5 | 29.5-34.5 | 34.5-39.5 | 39.5-44.5 | 44.5-49.5 |

Number of Tax Payers | 45 | 130 | 200 | 65 | 45 | 15 |

The histogram is shown below:

### Pie Diagram

A pie diagram is a very popular visual representation in business reports when a manager wants to show the share of various categories in total. The total is represented as a circle. Each category is depicted as a sector with its central angle proportional to its share. The share percent in total of each category is converted to a sector angle using the formula:

Sector Angle in degrees =

Other variations of pie diagrams are doughnut diagrams and exploded pie diagrams. These are shown below.

Example: ABC Company has a total income of ₹180 crores. Out of this, it has paid ₹10 crore as interest on borrowed capital. It has spent ₹80 crores on raw materials and other running expenditures. Its fixed costs (overheads) are ₹30 crores. On the net profit, it has to pay the tax at the rate of 30% on net profit.

Further, the board of directors decides to pay the dividend at the rate of 50% on the paid-up capital of ₹60 crore. The remaining amount is retained as profit plowed back. Depict the data as a pie diagram, doughnut diagram, and exploded pie diagram.

Amount in ₹ in Crore | Proportion to Total Income | Equivalent Angle | ||
---|---|---|---|---|

Total Income | (a) | 180 | 1 | 360 |

Expenditure on Raw Material | (b) | 80 | 0.44 | 160 |

Interest on Borrowed Capital | (c) | 10 | 0.056 | 20 |

Fixed Expenditure | (d) | 30 | 0.167 | 60 |

Net Profit [a – b – c –d] | (e) | 60 | ||

Tax [ 30 100 × e] | (f) | 18 | 0.1 | 36 |

Dividend | (g) | 30 | 0.167 | 60 |

Ploughed Back Capital | (h) | 12 | 0.167 | 24 |

**Solution**: We need to calculate the proportion of each category of the income distribution. Then we convert it to degrees, with the total being 360 degrees. The calculations are shown as follows:

### Frequency Polygon

A frequency polygon is used for presenting the frequency distribution in graphical form. This can be used for discrete distribution with grouped as well as ungrouped data. This can also be used for continuous data by converting it to approximate discrete data through grouping.

In all these cases, values of variables are represented on the X axis and their frequency (number of occurrences) on the Y axis. In the case of probability distributions, we use probability as frequency by choosing a suitable scale on the Y-axis.

For plotting the frequency polygon, we need to choose the appropriate scale and origin so that the main data features occupy a reasonable area on the paper. This helps readability. Although usually, the scale chosen is linear, however, depending on the data type we could use logarithmic or other types of scale.

Examples of these are audio noise plots, earthquake intensity plots, etc. Once the scale and origin are chosen, we need to draw grid lines (or use graph paper with grid lines) to facilitate accurate plotting.

Then we take each data point and mark it on the graph. In the case of grouped data, we use class marks (midpoints of the class intervals) as variable values on the X-axis.

These data points are joined by straight lines or a smooth curve to get a frequency polygon or frequency distribution in graphical form. To plot frequency distribution we can also join the data points by smooth lines.

**Example**: In a city, the income tax department had the data as follows for the number of taxpayers along with the range of income tax they paid for a particular year. Represent the data graphically with the help of a frequency polygon and frequency distribution chart.

Tax paid in ₹ ‘000 | 20-24 | 25-29 | 30-34 | 35-39 | 40-44 | 45-49 | Total |

Number of Taxpayers | 45 | 130 | 200 | 65 | 45 | 15 | 500 |

**Solution**: For plotting the data we will use class marks of tax-paid classes on the X axis and the number of taxpayers on the Y axis. Thus, the points for plotting are as follows. Then we join these points in straight lines.

Value on X axis | 22 | 27 | 32 | 37 | 42 | 47 |

Value on Y axis | 45 | 130 | 200 | 65 | 45 | 15 |

The plot is shown below:

To draw the plot as frequency distribution, we follow the same procedure for plotting the data points. Then we join the data points with a smooth curve as shown below. This gives better interpolation results. It also helps in comparing it with standard distributions.

### Ogives

Ogives are used to present the cumulative frequency of distribution in graphical format. There are two kinds of ogives. ‘Less than’ ogive represents cumulative frequency just below the variable value plotted on X axis.

On the other hand, ‘More than’ ogive plots the sum of the frequencies corresponding to above the variable value. For this, we first calculate ‘Less than’ and ‘More than’ cumulative frequencies for the entire variable values (corresponding to classes).

Then we plot these as points on the graph with class marks along the X axis and cumulative frequencies (‘Less than’ or ‘More than’) along the Y axis. These points are then joined by a smooth curve-like frequency distribution.

The value of the variable (on the X axis) at an ordinate from the point where two ogives intersect is ‘Median’ i.e. mid-value of the data (more about Median is in next chapter). The following example demonstrates the drawing of ogives.

**Example**: Before constructing a dam on a river the central water research institute performed a series of tests to measure the water flow, past the proposed location of the dam during the period of 246 days, when there was a sufficient flow of water. The results of the testing were used to construct the following frequency distribution.

River Flow (thousand cubic meters per min) | 1001-1050 | 1051-1100 | 1100-1150 | 1151-1200 | 1201-1250 | 1251-1300 | 1301-1350 | 1351-1400 |

Number of Days (frequency) | 7 | 21 | 32 | 49 | 58 | 41 | 27 | 11 |

- Draw ogive curves for the above data.
- From the ogive curve estimate the proportion of the days on which flow occurs at less than 1300 thousand cubic meters per minute

Solution: First we calculate and prepare the ‘less than’ and ‘more than’ frequency tables as follows.

River Flow 1000 cu. m per min | No of Days | Upper Class Limit | ‘Less than’ Frequency | Lower Class Limit | ‘More than’Frequency |
---|---|---|---|---|---|

1001 – 1050 | 7 | 1050.5 | 7 | 1001.5 | 246 |

1051 – 1100 | 21 | 1100.5 | 28 | 1050.5 | 239 |

1101 – 1150 | 32 | 1150.5 | 60 | 1100.5 | 218 |

1151 – 1200 | 49 | 1200.5 | 109 | 1150.5 | 186 |

1201 – 1250 | 58 | 1250.5 | 167 | 1200.5 | 137 |

1251 – 1300 | 41 | 1300.5 | 208 | 1250.5 | 79 |

1301 – 1350 | 27 | 1350.5 | 235 | 1300.5 | 38 |

1351 – 1400 | 11 | 1400.5 | 246 | 1350.5 | 11 |

Total | 246 |

Now we plot the ogives with class limits on the X axis and frequencies (less than or more than) on the Y axis and join the points with smooth curves. We also plot both ogives superimposed. The ogives are shown below.

From the ‘less than’ ogive we can read that the number of days on which flow occurs at less than 1,300 thousand cubic meters per minute is 208.

Thus, the proportion of days on which flow occurs at less than 1,300 thousand cubic meters per minute is 0.846 or 84.6%.