Graphical Representation of Data and Central Tendency

Statistics is a branch of mathematics that deals with methods for gathering, organising, and evaluating data such that useful conclusions can be drawn.

• Accidents, pricing of commodities, business, earnings, epidemics, sports data, and demographic statistics are examples of events that are dealt with.
• Differential and integral calculus, linear algebra, and probability theory are all used substantially in statistics' mathematical theories.
• By creating specialised experimental designs and survey samples, statistics can be utilised to improve data quality.
• Statistics also includes tools for forecasting and prediction.
• Statistics are useful in a wide range of academic fields, including natural and social sciences, government, and business.

Read Also: Equally Likely Events

Key Takeaways: Calculus, Integrals, Linear Algebra, Probability, Differentiation, Mean, Median, Mode, Statistics

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Introduction to Statistics

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Statistics is a field of study that deals with data gathering, presentation, interpretation, and analysis.

• Primary and Secondary Data: The data obtained is referred to as primary data when it was collected by the investigator herself or himself with a specific goal in mind. The data obtained is referred to as secondary data when it was gathered from a source that already had the information stored.
• Frequency: In statistics, frequency refers to the number of times a specific event occurs.
• Grouped data & Ungrouped data: Observations are sorted into groups in grouped data. Ungrouped data is data in its most basic or unprocessed form. There are no categories for the observations.
• Class Interval: The number of classes into which a set of data is divided. E.g., divisions on a histogram or bar graph. Class width = upper-class limit – lower class limit
• Regular and Irregular class interval: When the class intervals are identical or of the same size, it is called a regular class interval. For example, 0-10, 10-20, 20-30, 90-100. When the class intervals are of variable sizes, it is called an irregular class interval. For example, 0-35, 35-45, 45-55, 55-80, 80-90, 90-95, and 95-100.
• Frequency table: A frequency table, also known as frequency distribution, is a table that displays the frequency of a specific variable in a tabular format.
• Grouped frequency table & Ungrouped frequency table: The relevant class intervals' frequencies are organized or structured in a specific order, either ascending or descending. When each class interval's frequency is not ordered or organised in any way.

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Principles of Graphical Representation of Data

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Algebraic principles govern graphical representation. Two lines in a graph are known as the Axis or Coordinate axis. The X-axis and Y-axis are the two axes. The X-axis is the horizontal axis, while the Y-axis is the vertical axis. They are perpendicular to each other and cross at O, the origin point. The X-axis has a positive value on the right side of the Origin and a negative value on the left side. Similarly, the positive side of the Origin Y-axis has a positive value, whereas the negative side has a negative value. When the x- and y-axis connect at the origin, the plane is divided into four quadrants: Quadrant I, Quadrant II, Quadrant III, and Quadrant IV. The Histogram, Smoothed frequency graph, Pie diagram or Pie chart, Cumulative or ogive frequency graph, and Frequency Polygon are examples of this type of representation. The following are the main graphical representations:

Bar Graph

A bar graph is a visual representation of data in which bars of uniform width are drawn on one axis (say, the x-axis) with equal spacing between them, displaying the variable. The variable's values are displayed on the other axis (say, the y-axis), and the heights of the bars are determined by the variable's values.

Check Important Difference Between Average and Mean

The data can be represented as:

Histogram

A histogram is a graphical depiction of a continuous-classed grouped frequency distribution. The construction steps are as follows:

• On a suitable scale, we represent the class limits along the x-axis and the frequencies along the y-axis.
• We now create rectangles (or rectangular bars) of equal width and length based on the frequencies of the corresponding class intervals.

The generated graph resembles a solid figure since there are no gaps between consecutive rectangles. This is referred to as a histogram.

Note:

• If the first-class interval does not begin at zero, we mark a kink or a break on the axis to represent it on the graph.
• The area of the rectangles erected in a histogram is proportional to the associated frequencies.
• If the widths of the rectangles vary, we must make changes to the lengths of the rectangles so that the areas are proportionate to the frequencies once again. The following are the measures to take to do this.
• Choose a class interval with the smallest possible class size.
•  The rectangles' lengths are then adjusted to be proportional to the minimal class size. 'Lengths proportional to the minimal class size' is how these lengths are referred to.

Presentation of Histogram is as follows:

Frequency polygon

The frequency polygon is created for an ungrouped distribution by plotting points with the abscissa as the variate values and the ordinate as the associated frequencies, and then connecting the plotted points with straight lines. The abscissa of points in a grouped frequency distribution is the mid-values of the class intervals. The frequency polygon can be made for equal class intervals by connecting the middle points of the upper sides of the neighbouring rectangles of the histogram with straight lines.

Note: If a histogram and a frequency polygon are to be drawn, the histogram should be drawn first, followed by the frequency polygon.

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Rules of Graphical Representation of Data

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There are some rules that must be followed when presenting data graphically. They are as follows:

• Suitable Title: The graph's title should be appropriate and identify the presentation's topic.
• Measurement Unit: The graph's measurement unit should be specified.
• Proper Scale: To accurately depict the data, a proper scale must be chosen.
• Index the relevant hues, tones, lines, and designs in the graphs for better understanding.
• Data Sources: At the bottom of the graph, data should be placed wherever it is needed.
• Simple: A graph's construction should be simple to comprehend.
• To interpret the data effectively, the graph should be visually clean in terms of size and typeface.

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Measures of Central Tendency

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There are three main averages:

Mean

The sum of all the values of all the observations divided by the total number of observations is the mean (or average) of a set of observations. Thus,

In general, for n observations,

For an ungrouped frequency distribution,

Where, fi is the frequency corresponding to the observation xi.

Median

The median is the value that splits a given number of observations into exactly two halves. When the observations are sorted in ascending or descending order of magnitudes, the median is the middle or centre value of the variate in a set of observations.

Median is calculated as: ½ (n+1), where n is the number of values in the data.

The median is the average of the two middle values if the number of items in the data collection is even.

Mode

The mode is the value of an observation that happens the most frequently, i.e., the observation with the highest frequency. The data's extreme values have an impact on the mean. Extreme numbers in the data do not affect the median or mode.

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Points to Remember

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Following are the important points:

• The most common observation is referred to as the mode.
• The modal class is the class interval with the highest frequency.
• Raw data needs to be sorted to carry out operations. This process is known as sorting.
• The class mark of a class is the mid-value of the two limits of that class.
• An Inclusive or Discontinuous Frequency Distribution is a frequency distribution in which the upper limit of one class differs from the lower limit of the next class.

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