Median of Grouped Data

Definition: The median of a data set is the middle value when all observations are arranged in ascending order. For grouped data, it is the value below which 50% of the total observations fall.

Key terms:

• Cumulative frequency (cf) -- the running total of frequencies up to and including each class.
• n -- total number of observations = Sigma(f_i).
• n/2 -- half the total frequency (the position of the median).
• Median class -- the class interval whose cumulative frequency is the first to exceed n/2.
• l -- lower limit of the median class.
• f -- frequency of the median class.
• cf -- cumulative frequency of the class just BEFORE the median class.
• h -- class size (width) of the median class.

Derivation of the Median Formula:

1. The median is the (n/2)-th observation.
2. From the cumulative frequency table, the median lies in the class whose cumulative frequency first reaches or exceeds n/2.
3. Let this be the class [l, l + h) with frequency f.
4. The cumulative frequency up to the class before the median class is cf.
5. So (n/2 - cf) observations within the median class need to be counted to reach the median.
6. Assuming observations are uniformly distributed within each class:
7. The fraction of the class to be covered = (n/2 - cf)/f.
8. The median is located at a distance of [(n/2 - cf)/f] x h from the lower limit l.
9. Median = l + [(n/2 - cf)/f] x h. Hence derived.

Assumption: The formula assumes that observations within each class interval are uniformly (evenly) distributed. This is a reasonable approximation for most practical data sets.

Problems on median of grouped data include:

Type 1: Direct Calculation

• Given a frequency distribution, find the median using the formula.

Type 2: Finding Missing Frequency

• Given the median and all but one frequency, find the missing frequency.

Type 3: Median from Cumulative Frequency Table

• Data is given in "less than" or "more than" cumulative form. Convert to class intervals and find the median.

Type 4: Ogive-Based Median

• Draw the ogive (cumulative frequency curve), mark n/2 on the y-axis, draw a horizontal line to the curve, and drop a perpendicular to the x-axis to read the median.

Type 5: Comparing Mean, Median, and Mode

• Calculate all three measures for the same data set and compare.

Step-by-step method to find median of grouped data:

1. Construct the cumulative frequency table.
2. Find n/2 where n = Sigma(f_i).
3. Identify the median class -- the class whose cumulative frequency first exceeds n/2.
4. Note the values:
   • l = lower limit of median class
   • f = frequency of median class
   • cf = cumulative frequency of the class just before the median class
   • h = class size
5. Apply the formula: Median = l + [(n/2 - cf)/f] x h.
6. Simplify to get the answer.

Finding median from an Ogive:

1. Plot the ogive (cumulative frequency vs. upper class boundary).
2. Mark n/2 on the y-axis.
3. Draw a horizontal line from n/2 to the ogive curve.
4. From the point of intersection, drop a perpendicular to the x-axis.
5. The x-coordinate of this point is the median.

Common Mistakes:

• Using the wrong cf -- it must be the cumulative frequency of the class BEFORE the median class, not the median class itself.
• Confusing n/2 with (n+1)/2. For grouped data, always use n/2.
• Taking the upper limit instead of the lower limit for l.

Economics:

• Median income is more meaningful than mean income because it is not distorted by extremely high earners.

Real Estate:

• Median house prices are reported instead of mean prices to avoid distortion from very expensive properties.

Education:

• Median marks give a better sense of typical performance than the mean, especially with extreme scores.

Government:

• Poverty line calculations, median household size, and median age are used for policy planning.

Healthcare:

• Median survival time in clinical trials is preferred over mean because it is unaffected by outliers.