We have learnt that to represent the data pictorially in the form of various graphs such as bar graphs, histograms (including those of varying widths) and frequency polygons and also numerical representatives of the ungrouped data, also called measures of central tendency, namely, mean, median and mode.
We will learn the concept of cumulative frequency, the cumulative frequency distribution and how to draw cumulative frequency curves, called ogives.
As we had learn the mean (or average) of observations, the sum of the values of all the observations divided by the total number of observations.
The mean x of the data is given by
x = f1x1 + f2x2 + ... + fnxn/f1 + f2 + ... + fn
in short form by using the Greek letter S (capital sigma) which means summation. That is,
x = Σni=1fixi/Σni=1fi
It is assumed that the frequency of each class interval is centered on its mid-point. So the mid-point (or class mark) of each class can be chosen to represent the observations falling in the class. Recall that we find the mid-point of a class (or its class mark) by finding the average of its upper and lower limits. That is,
A mode is the value among the observations which occurs most often, that's the value of the observation having the maximum frequency.
Here we will learn the ways of obtaining a mode of grouped data.
Mode = l + (f1 - f0/2f1-f0 - f2) x h
Where l = lower limit of the modal class,
h = size of the class interval (assuming all class sizes to be equal),
f1 = frequency of the modal class,
f0 = frequency of the class preceding the modal class,
f2 = frequency of the class succeeding the modal class.
Median is a measurement of central tendency which gives the value of the middle-most observation in the data.
Median = l + (n/2-cf/100) x h
Where l = lower limit of median class,
n = number of observations,
cf = cumulative frequency of class preceding the median class,
f = frequency of median class,
h = class size (assuming class size to be equal).
3 Median = Mode + 2 Mean
A graphical representation helps us =in understanding given data at a glance.
We have learnt to represent the data through bar graphs, histograms and frequency polygons.
Let us hear learn to represent a cumulative frequency distribution graphically.
The term ‘ogive’ is pronounced as ‘ojeev’ and is derived from the word ogee. An ogee is a shape consisting of a concave arc flowing into a convex arc, so forming an S-shaped curve with vertical ends.
To represent ‘the more than type’ graphically, we plot the lower limits on the x-axis and the corresponding cumulative frequencies on the y-axis. Then we plot the points (lower limit, corresponding cumulative frequency), i.e., (0, 53), (10, 48), (20, 45), (30, 41), (40, 38), (50, 35), (60, 31), (70, 24), (80, 15), (90, 8), on a graph paper, and join them by a free hand smooth curve.
The curve we get is a cumulative frequency curve, or an ogive (of the more than type).
