Statistics: Frequency Distributions and Graphical Representation

Data refers to a collection of facts, such as numbers, words, measurements, observations, or even just descriptions of things. When collected in its original form, it is called raw data. Data can be broadly classified into two types:\n1. Primary Data: Data collected by the investigator himself for a definite purpose or inquiry.\n2. Secondary Data: Data collected by someone else and used by the investigator for his purpose.

The frequency of an observation is the number of times that particular observation occurs in a given set of data. For example, if the number '5' appears 7 times in a data set, then the frequency of '5' is 7.

A frequency distribution is a tabular summary of data showing the frequency of each value or class interval. It helps in organising raw data into a more meaningful form.\n\n1. Ungrouped Frequency Distribution: When the number of observations is small and the range of data is not very wide, we can prepare an ungrouped frequency distribution table. Each distinct observation is listed, and its frequency is recorded.\n\n2. Grouped Frequency Distribution: When the number of observations is large and the range of data is wide, it is convenient to group the data into class intervals. This is called a grouped frequency distribution.\n * Class Interval: The range of values in each group (e.g., 0-10, 10-20).\n * Class Size (or Class Width): The difference between the upper and lower class limits of a class interval. For example, for the interval 10-20, the class size is 20 - 10 = 10.\n * Lower Class Limit: The smallest value in a class interval.\n * Upper Class Limit: The largest value in a class interval.\n * Class Mark (or Mid-point): The mid-point of a class interval. It is calculated as (Lower Class Limit + Upper Class Limit) / 2.

A bar graph is a pictorial representation of data in which bars of uniform width are drawn with equal spacing between them on one axis (usually the x-axis) and the heights of the bars represent the frequency of the respective observations on the other axis (usually the y-axis). Bar graphs are generally used for discrete data or categorical data.

A histogram is a graphical representation of a grouped frequency distribution with continuous classes. In a histogram, the class intervals are taken on the x-axis and their corresponding frequencies on the y-axis. Rectangles are constructed with bases as the class intervals and heights proportional to the frequencies. There are no gaps between adjacent bars in a histogram because the data is continuous.\n\nFor unequal class widths, the height of the rectangle is adjusted by calculating the 'frequency density' for each class. Frequency density = Frequency / Class Width. The area of each rectangle is then proportional to the frequency.

A frequency polygon is another way of representing a frequency distribution graphically. It can be drawn in two ways:\n1. By joining the mid-points of the tops of the adjacent rectangles of a histogram with line segments.\n2. By plotting the class marks against the frequencies and then joining these points with line segments. To make it a closed figure, the mid-points of the immediately lower and higher hypothetical class intervals (with zero frequency) are also included and connected to the first and last plotted points, respectively.