Ogives and the Empirical Relationship

Two Kinds of Cumulative Frequency Distribution

A cumulative frequency distribution can be written two ways:

• Less-than type: for each class, how many observations are less than its upper limit. The cumulative counts increase down the table.
• More-than type: for each class, how many observations are greater than or equal to its lower limit. These counts decrease down the table.

Both describe the same data — just accumulated from opposite ends.

Drawing an Ogive

An ogive (cumulative frequency curve) is the graph of a cumulative frequency distribution — a smooth free-hand curve, not straight-line bars.

• Less-than ogive: plot the points (upper class limit, cumulative frequency) and join them with a smooth rising curve.
• More-than ogive: plot the points (lower class limit, cumulative frequency) and join them with a smooth falling curve.

Key Point: Less-than $\to$ upper limits, rising curve. More-than $\to$ lower limits, falling curve.

Finding the Median from an Ogive

There are two graphical ways to read the median:

1. From one ogive. Mark $\dfrac{n}{2}$ on the cumulative-frequency (vertical) axis, draw a horizontal line to the less-than ogive, then drop a vertical line to the horizontal axis — that x-value is the median.

2. From both ogives. Draw the less-than and more-than ogives on the same axes. They cross at one point; the x-coordinate of that intersection is the median.

Key Point: The two ogives intersect at the median; equivalently, the less-than ogive reaches the median at height $\dfrac{n}{2}$.

The Empirical Relationship

For a moderately (not too heavily) skewed distribution, the three measures are connected by an empirical relationship: $$\boxed{3\,\text{Median} = \text{Mode} + 2\,\text{Mean}}$$

Rearranged, it lets you estimate any one from the other two: $$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}, \qquad \text{Mean} = \dfrac{3\,\text{Median} - \text{Mode}}{2}, \qquad \text{Median} = \dfrac{\text{Mode} + 2\,\text{Mean}}{3}.$$

[Board Important] This is approximate — use it only when the question asks you to estimate the third measure from the other two.

Solved Examples

Example 1: Empirical relation — find the mode

The mean of a distribution is 45 and its median is 48. Estimate the mode.

Solution:

1. Mode $= 3\,\text{Median} - 2\,\text{Mean} = 3(48) - 2(45) = 144 - 90 = 54$.

Final Answer: Mode $\approx 54$.

Takeaway: $\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$.

Example 2: Empirical relation — find the median

The mode of a distribution is 25 and its mean is 28. Estimate the median.

Solution:

1. $\text{Median} = \dfrac{\text{Mode} + 2\,\text{Mean}}{3} = \dfrac{25 + 56}{3} = \dfrac{81}{3} = 27$.

Final Answer: Median $\approx 27$.

Takeaway: Rearrange $3\,\text{Median}=\text{Mode}+2\,\text{Mean}$.

Example 3: Building 'less than' points

From frequencies $5, 8, 12, 10, 5$ (classes $0\text{-}10,\dots,40\text{-}50$), list the points for a less-than ogive.

Solution:

1. cf: $5, 13, 25, 35, 40$.
2. Less-than points (upper limit, cf): $(10,5), (20,13), (30,25), (40,35), (50,40)$.

Final Answer: $(10,5),(20,13),(30,25),(40,35),(50,40)$.

Takeaway: Plot cumulative frequency against the upper limits and join smoothly.

Example 4: Median from $\tfrac n2$ on the ogive

For the data of Example 3 ($n=40$), where would you read the median on the less-than ogive?

Solution:

1. $\dfrac{n}{2} = 20$. Draw a horizontal line from 20 on the cf-axis to the curve and drop to the x-axis.
2. Between $(20,13)$ and $(30,25)$ the curve reaches cf $20$; by the median formula this is $10 + \dfrac{20-13}{12}\times 10 = 15.83$.

Final Answer: Median $\approx 15.83$ (read where the ogive is at height 20).

Takeaway: The ogive reading and the median formula agree.