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| Graphical Representation of Cumulative Frequency Distribution (Ogive) | Types of Ogives (Less Than Ogive, More Than Ogive) | Estimation of Median (Graphically from Ogives) |
Graphical Representation: Cumulative Frequency Graphs
Graphical Representation of Cumulative Frequency Distribution (Ogive)
Definition and Purpose
An Ogive (pronounced "oh-jive" and sometimes called a cumulative frequency curve) is a specific type of line graph used to represent a cumulative frequency distribution. It plots the cumulative frequencies against the respective upper or lower boundaries of the class intervals.
Ogives are instrumental in statistics as they allow for the visual determination of percentiles and provide a smooth curve representation of how data accumulates over the range of values. They are particularly useful for:
- Easily finding the number or proportion of observations that fall below or above a certain value.
- Graphically estimating partition values like the median, quartiles (Q1, Q2, Q3), deciles (D1 to D9), and percentiles (P1 to P99).
- Providing a continuous visual representation of the cumulative distribution of a dataset.
Unlike histograms and frequency polygons which show frequencies within intervals, an ogive shows the running total of frequencies.
Construction Principles of an Ogive
Constructing an ogive requires first preparing a cumulative frequency table. The choice of points to plot depends on whether you are drawing a 'less than' or a 'more than' ogive.
-
Prepare the Cumulative Frequency Table:
Start with a grouped frequency distribution table. Calculate either the 'less than' cumulative frequencies or the 'more than' cumulative frequencies (or both if you plan to draw both ogives on the same graph).
- For 'Less Than' CF: Sum frequencies starting from the lowest class upwards. The cumulative frequency for a class is the sum of its frequency and the frequencies of all preceding classes. It is associated with the upper boundary of the class interval.
- For 'More Than' CF: Sum frequencies starting from the highest class downwards, or subtract frequencies from the total frequency. The cumulative frequency for a class is the sum of its frequency and the frequencies of all succeeding classes. It is associated with the lower boundary of the class interval.
-
Identify Points to Plot:
Based on the type of ogive, select the points that will be plotted on the graph paper:
- For a 'Less Than' Ogive: The points are (Upper Class Boundary, 'Less Than' Cumulative Frequency).
To start the curve from the x-axis, include a point corresponding to the lower boundary of the first class with a cumulative frequency of 0. For example, if the first class is 40-45, the first point is (40, 0).
- For a 'More Than' Ogive: The points are (Lower Class Boundary, 'More Than' Cumulative Frequency).
To end the curve on the x-axis, include a point corresponding to the upper boundary of the last class with a cumulative frequency of 0. For example, if the last class is 70-75, the last point is (75, 0).
Ensure you use the actual class boundaries, especially if the original frequency distribution uses inclusive intervals. For example, if the inclusive class is 10-19, the boundaries are 9.5-19.5. The 'less than' ogive point would be (19.5, cf), and the 'more than' ogive point would be (9.5, cf).
- For a 'Less Than' Ogive: The points are (Upper Class Boundary, 'Less Than' Cumulative Frequency).
-
Draw and Label Axes:
Draw two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Label the x-axis with the variable being measured (e.g., Weight in kg, Marks). Mark the class boundaries on this axis.
Label the y-axis as "Cumulative Frequency". The scale on the y-axis should range from 0 up to the total frequency ($N$). Choose equal increments along both axes.
-
Plot and Connect Points:
Plot the points identified in Step 2 on the graph paper according to the chosen scales. Connect these plotted points consecutively with smooth curves or straight line segments. A smooth curve is often preferred as it better represents the continuous nature of cumulative data.
-
Add a Title:
Give the ogive a clear and descriptive title, specifying whether it is a 'less than' or 'more than' ogive (e.g., "Less Than Ogive of Student Weights").
Types of Ogives (Less Than Ogive, More Than Ogive)
Based on the type of cumulative frequency used, there are two main types of ogives:
1. Less Than Ogive
A 'less than' ogive is constructed using the 'less than' cumulative frequency distribution. It shows the total number of observations that are less than the upper boundary of each class interval.
- Construction: Plots the upper class boundaries on the x-axis against the corresponding 'less than' cumulative frequencies on the y-axis.
- Starting Point: The curve starts from the lower boundary of the first class interval on the x-axis, where the cumulative frequency is 0.
- Shape: The 'less than' ogive is always a non-decreasing curve (it either rises or stays flat, never falls). It typically starts at 0 on the y-axis and ends at the total frequency ($N$) at the upper boundary of the last class interval. It often takes on an elongated S-shape.
- Interpretation: If you locate a value $x$ on the x-axis and find the corresponding point $(x, y)$ on the 'less than' ogive, the value $y$ on the y-axis tells you that there are $y$ observations with values less than $x$.
2. More Than Ogive
A 'more than' ogive is constructed using the 'more than' cumulative frequency distribution. It shows the total number of observations that are greater than or equal to the lower boundary of each class interval.
- Construction: Plots the lower class boundaries on the x-axis against the corresponding 'more than' cumulative frequencies on the y-axis.
- Starting Point: The curve starts from the lower boundary of the first class interval on the x-axis, where the cumulative frequency is equal to the total frequency ($N$), as all observations are greater than or equal to this boundary.
- Shape: The 'more than' ogive is always a non-increasing curve (it either falls or stays flat, never rises). It typically starts at the total frequency ($N$) on the y-axis and ends at 0 on the y-axis at the upper boundary of the last class interval. It often takes on an inverted S-shape.
- Interpretation: If you locate a value $x$ on the x-axis and find the corresponding point $(x, y)$ on the 'more than' ogive, the value $y$ on the y-axis tells you that there are $y$ observations with values greater than or equal to $x$.
Example
Example 1. Draw the 'Less Than' Ogive and the 'More Than' Ogive for the following frequency distribution of student weights:
| Weight (kg) | Frequency (f) |
|---|---|
| 40 - 45 | 2 |
| 45 - 50 | 5 |
| 50 - 55 | 5 |
| 55 - 60 | 7 |
| 60 - 65 | 6 |
| 65 - 70 | 4 |
| 70 - 75 | 1 |
| Total | 30 |
Answer:
Given: Grouped frequency distribution table for student weights.
To Construct: 'Less Than' and 'More Than' Ogives.
Solution:
First, we prepare the cumulative frequency tables.
Less Than Cumulative Frequency Table:
| Weight (kg) (Less than Upper Boundary) |
Frequency (f) | Less Than Cumulative Frequency (cf) |
|---|---|---|
| Less than 40 | 0 | 0 |
| Less than 45 | 2 | 2 |
| Less than 50 | 5 | $2 + 5 = 7$ |
| Less than 55 | 5 | $7 + 5 = 12$ |
| Less than 60 | 7 | $12 + 7 = 19$ |
| Less than 65 | 6 | $19 + 6 = 25$ |
| Less than 70 | 4 | $25 + 4 = 29$ |
| Less than 75 | 1 | $29 + 1 = 30$ |
| Total | 30 |
Points for 'Less Than' Ogive: (40, 0), (45, 2), (50, 7), (55, 12), (60, 19), (65, 25), (70, 29), (75, 30).
More Than Cumulative Frequency Table:
| Weight (kg) (More than or equal to Lower Boundary) |
Frequency (f) | More Than Cumulative Frequency |
|---|---|---|
| More than or equal to 40 | 2 | 30 |
| More than or equal to 45 | 5 | $30 - 2 = 28$ |
| More than or equal to 50 | 5 | $28 - 5 = 23$ |
| More than or equal to 55 | 7 | $23 - 5 = 18$ |
| More than or equal to 60 | 6 | $18 - 7 = 11$ |
| More than or equal to 65 | 4 | $11 - 6 = 5$ |
| More than or equal to 70 | 1 | $5 - 4 = 1$ |
| More than or equal to 75 | 0 | 0 |
Points for 'More Than' Ogive: (40, 30), (45, 28), (50, 23), (55, 18), (60, 11), (65, 5), (70, 1), (75, 0).
Now, we plot these points on a graph paper with Weight on the x-axis and Cumulative Frequency on the y-axis (from 0 to 30).
Title: Less Than and More Than Ogives for Student Weights
Estimation of Median (Graphically from Ogives)
The median is a measure of central tendency that represents the middle value in a dataset when the data is arranged in ascending or descending order. In a frequency distribution, the median is the value below which 50% of the observations lie. Ogives provide a very convenient graphical method to estimate the median of a grouped frequency distribution.
Method 1: Using the 'Less Than' Ogive
The 'less than' ogive shows the number of observations below a certain value. To find the median graphically using this ogive, we find the value on the x-axis below which half of the total observations lie.
- First, draw the 'Less Than' Ogive for the given frequency distribution.
- Calculate the position of the median observation. This is given by $\frac{N}{2}$, where $N$ is the total frequency.
- Locate the value $\frac{N}{2}$ on the vertical axis (the cumulative frequency axis).
- Draw a horizontal line segment from the point corresponding to $\frac{N}{2}$ on the y-axis, extending it towards the right until it intersects the 'Less Than' Ogive curve.
- From the point where the horizontal line intersects the ogive curve, draw a vertical line segment downwards, perpendicular to the horizontal axis.
- The value on the horizontal axis where this vertical line intersects is the estimated Median of the distribution.
Method 2: Using Both Ogives
Drawing both the 'Less Than' and 'More Than' ogives on the same graph provides an alternative and often more precise graphical method for estimating the median.
- Draw both the 'Less Than' Ogive and the 'More Than' Ogive on the same graph paper, using the same scales for both axes.
- Locate the point where the two ogive curves intersect. This intersection point represents the value on the x-axis where the number of observations less than that value equals the number of observations greater than or equal to that value (i.e., the middle value).
- From the point of intersection of the two ogives, draw a vertical line segment downwards, perpendicular to the horizontal axis.
- The value on the horizontal axis where this vertical line intersects is the estimated Median of the distribution.
When both ogives are drawn correctly on the same graph, their intersection point's y-coordinate will always be equal to $\frac{N}{2}$. This provides a visual check for the calculation of $\frac{N}{2}$.
It is important to note that estimation from a graph depends on the accuracy of the drawing and the chosen scale. For a more precise median value, the formula for calculating the median of grouped data should be used.
Example
Example 1. Estimate the median weight of the students graphically using the 'Less Than' Ogive from the previous example (Total frequency $N=30$).
Answer:
Given: 'Less Than' Ogive and total frequency $N=30$.
To Estimate: The median weight graphically.
Solution:
We will use Method 1 (using the 'Less Than' Ogive).
- The 'Less Than' Ogive for the student weights is already constructed as shown in Example 1 of section I2.
- Calculate the position of the median observation:
$\frac{N}{2} = \frac{30}{2} = 15$
This means the median weight is the value on the x-axis corresponding to a cumulative frequency of 15.
- Locate the value 15 on the vertical axis (Cumulative Frequency).
- Draw a horizontal line from the point $15$ on the y-axis to the right, until it intersects the 'Less Than' Ogive curve.
- From the point of intersection on the curve, draw a vertical line straight down to the horizontal axis (Weight).
- Read the value on the horizontal axis where the vertical line intersects. Looking at the graph from Example 1 in the previous section, the horizontal line from CF = 15 intersects the curve between the points (55, 12) and (60, 19). The vertical line appears to intersect the x-axis somewhere within the 55-60 kg interval. By visual estimation from a carefully drawn graph:
The estimated value on the x-axis appears to be approximately 57.1 or 57.2 kg.
The estimated median weight from the graph is approximately 57.1 kg (this value might vary slightly depending on the precision of the graph drawing and reading).
Alternative Method (Using Both Ogives - Conceptual):
If both the 'Less Than' and 'More Than' Ogives were drawn on the same graph (as shown in the example in section I2), we would find the point where they intersect. The x-coordinate of this intersection point would also provide the estimated median. Observing the graph in Example 1 of section I2, the intersection appears to occur at a weight value between 55 and 60 kg, and at a cumulative frequency of 15 (the calculated $\frac{N}{2}$). The x-value at this intersection visually corresponds to the estimated median weight.