Measures of Central Tendency: Median

Median: Definition and Calculation for Ungrouped Data

Definition and Properties

The Median is a measure of central tendency that represents the middle value in a dataset that has been arranged in order, either ascending (from smallest to largest) or descending (from largest to smallest). It is a positional average because its value depends on its position in the ordered dataset rather than on the magnitude of every observation.

Key properties of the median:

• It divides the data into two equal halves: approximately 50% of the observations are less than or equal to the median, and 50% are greater than or equal to the median.
• It is less affected by extreme values (outliers) compared to the mean. If a dataset has a few unusually large or small values, the median will provide a more typical representation of the center than the mean.
• It can be calculated for ordinal, interval, and ratio scale data.
• It is not based on the magnitude of every observation in the dataset in its calculation formula (unlike the mean), although all observations must be considered for ordering the data.

Calculation for Ungrouped Data

Ungrouped data refers to raw data where observations are listed individually or are presented in an ungrouped frequency distribution (where individual values and their frequencies are given). The method for finding the median depends on whether the total number of observations is odd or even.

Case 1: When Individual Observations are Given

To find the median of a list of individual observations:

1. Arrange the Data: The first and most crucial step is to arrange the observations in either ascending (smallest to largest) or descending (largest to smallest) order. The result will be the same.
2. Count Observations: Determine the total number of observations, denoted by $n$.
3. Locate the Median Position and Value:
   • If $n$ is Odd: There is a single middle observation. The position of the median is given by the formula $\left(\frac{n+1}{2}\right)^{\text{th}}$. The median is the value of the observation found at this position in the ordered list.
   • If $n$ is Even: There are two middle observations. The positions of these two middle observations are $\left(\frac{n}{2}\right)^{\text{th}}$ and $\left(\frac{n}{2} + 1\right)^{\text{th}}$. The median is calculated as the arithmetic mean (average) of the values of these two middle observations.

     Median $= \frac{\text{Value of } \left(\frac{n}{2}\right)^{\text{th}} \text{ obs} + \text{ Value of } \left(\frac{n}{2} + 1\right)^{\text{th}} \text{ obs}}{2}$

     ... (1)

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Case 2: When Data is in an Ungrouped Frequency Distribution

If the data is given in an ungrouped frequency distribution table (distinct values with their frequencies), we can find the median by using cumulative frequencies.

1. Arrange Distinct Values: Ensure the distinct values ($x_i$) are listed in ascending order.
2. Calculate Cumulative Frequencies: Add a column for cumulative frequency (cf). The cumulative frequency for a value is the sum of its frequency and the frequencies of all preceding values.
3. Find Total Frequency: Calculate the total number of observations, $N = \sum f_i$.
4. Locate Median Position: The median position is at $\left(\frac{N+1}{2}\right)^{\text{th}}$ if considering the 'exact' middle observation rank, or more commonly $\frac{N}{2}$ is used to locate the class/value containing the median. Let's use the position $\frac{N}{2}$ for simplicity as it directly corresponds to the cumulative frequency definition.
5. Identify the Median Value: Find the value in the cumulative frequency column that is just greater than or equal to the median position ($\frac{N}{2}$). The distinct value ($x_i$) corresponding to this cumulative frequency is the median. (Note: If $\frac{N}{2}$ exactly matches a cumulative frequency, the median is the average of the corresponding $x_i$ and the next $x_i$, but for discrete data, the corresponding $x_i$ is often considered the median).

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Median of Grouped Data (Formula and Calculation)

Introduction and Median Class

When data is grouped into class intervals, we lose the exact values of individual observations. Therefore, we cannot find the precise median as we do with ungrouped data. Instead, we estimate the median based on the frequency distribution, assuming that observations are evenly distributed within the class interval containing the median. This interval is called the median class.

The median of grouped data is the value that divides the total frequency ($N$) into two equal halves. The median value lies within the median class.

Steps to Calculate the Median of Grouped Data

Follow these steps to calculate the median from a grouped frequency distribution table:

1. Calculate Cumulative Frequencies:

   Prepare a 'less than' cumulative frequency (cf) column for the frequency distribution. Sum frequencies from the first class downwards. Find the total frequency $N = \sum f_i$.

2. Identify the Median Class Position:

   Calculate $\frac{N}{2}$. This value represents the position of the median observation in the cumulative frequency distribution.

3. Locate the Median Class:

   Find the class interval in the frequency distribution whose 'less than' cumulative frequency is just greater than or equal to the value $\frac{N}{2}$. This class interval is the median class. The median of the data lies within this class.

4. Determine Values for the Formula:

   Once the median class is identified, extract the following values needed for the median formula:

   • $l$: The lower class boundary of the median class. (Use the lower limit if using exclusive intervals; convert inclusive limits to boundaries if necessary).
   • $N$: The total frequency.
   • $cf$: The cumulative frequency of the class immediately preceding the median class.
   • $f$: The frequency of the median class itself.
   • $h$: The class width (size) of the median class. (Assumed to be constant for most problems). $h = \text{Upper Boundary} - \text{Lower Boundary}$.

5. Apply the Median Formula:

   The median of grouped data is calculated using the formula:

   Median $= l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h$

   ... (1)

This formula interpolates within the median class, assuming a linear distribution of observations within that class.

Derivation of the Median Formula for Grouped Data

Let the median class be $C_m = [l, u)$, with frequency $f$. Let $cf$ be the cumulative frequency of the class just before $C_m$. The total frequency is $N$. The median is the value $M$ such that $\frac{N}{2}$ observations are less than $M$.

We know that $cf$ observations are less than $l$ (the lower boundary of the median class).

We need $\frac{N}{2}$ observations to be less than the median $M$. The number of observations needed from the median class to reach the median is $\frac{N}{2} - cf$.

Assuming observations are uniformly distributed within the median class $[l, u)$, the frequency $f$ is spread over the class width $h = u - l$.

This means $f$ observations are spread over a width of $h$.

So, 1 observation is spread over a width of $\frac{h}{f}$.

The number of observations needed from the median class is $\left(\frac{N}{2} - cf\right)$. The width covered by these observations within the median class is $\left(\frac{N}{2} - cf\right) \times \frac{h}{f}$.

The median value $M$ is the lower boundary of the median class ($l$) plus this additional width:

This formula provides an estimated median value for grouped data.

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Estimation of Median (Graphically from Ogives)

As discussed in the section on graphical representation of cumulative frequency distributions (Ogives), the median for grouped data can be estimated visually from an ogive (or the intersection of two ogives). This method provides a graphical approximation of the median value.

The median is the value on the horizontal axis corresponding to a cumulative frequency of $\frac{N}{2}$.

Method 1: Using the 'Less Than' Ogive

The 'less than' ogive plots upper class boundaries against 'less than' cumulative frequencies. To find the median graphically from this ogive:

1. Draw the 'Less Than' Ogive: Construct the 'less than' cumulative frequency table and plot the corresponding ogive curve on a graph paper, with upper class boundaries on the x-axis and cumulative frequencies on the y-axis.
2. Calculate Median Position: Determine the position of the median observation in the cumulative frequency distribution by calculating $\frac{N}{2}$, where $N$ is the total frequency.
3. Locate Position on Y-axis: Find the value $\frac{N}{2}$ on the vertical axis (the cumulative frequency axis).
4. Draw Horizontal Line: From the point on the y-axis corresponding to $\frac{N}{2}$, draw a horizontal line segment parallel to the x-axis, extending it towards the right until it intersects the 'Less Than' Ogive curve.
5. Draw Vertical Line: From the point where the horizontal line intersects the ogive curve, draw a vertical line segment downwards, perpendicular to the horizontal axis.
6. Read the Median Value: The value on the horizontal axis where this vertical line intersects is the estimated Median of the distribution.

Method 2: Using Both 'Less Than' and 'More Than' Ogives

When both types of ogives are drawn on the same graph, their intersection point provides a direct visual estimate of the median.

1. Draw Both Ogives: Construct both the 'Less Than' Ogive and the 'More Than' Ogive on the same graph paper, using the same scales for both axes. Ensure the 'less than' ogive starts at the lower boundary of the first class with CF=0 and the 'more than' ogive ends at the upper boundary of the last class with CF=0.
2. Find Intersection Point: Locate the point where the two ogive curves intersect each other.
3. Draw Vertical Line: From the point of intersection of the two ogives, draw a vertical line segment downwards, perpendicular to the horizontal axis.
4. Read the Median Value: The value on the horizontal axis where this vertical line intersects is the estimated Median of the distribution.

When drawn accurately, the y-coordinate of the intersection point of the two ogives will be equal to $\frac{N}{2}$. This method provides a visual cross-check and estimate of the median value.

These graphical methods are useful for quickly estimating the median and visualizing its position within the distribution. However, the formula method for grouped data provides a more precise calculated estimate based on specific assumptions about the data distribution within the median class.

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