Median of Ungrouped Data

Definition: The median of a data set is the value that divides the data into two equal halves when arranged in order. Half the observations lie below the median and half lie above it.

Median = Middle value after arranging data in ascending order

For n observations arranged in ascending order:

• If n is odd: Median = value of the ((n + 1)/2)th observation.
• If n is even: Median = average of the (n/2)th and ((n/2) + 1)th observations.

Important:

• The data MUST be arranged in ascending (or descending) order before finding the median.
• The median may or may not be a value that actually appears in the data set.
• For an odd number of observations, the median is one of the data values.
• For an even number of observations, the median is the average of two middle values and may not be a data value.

Why the Median Formula Works:

Case 1: Odd number of observations (n is odd)

1. If there are n observations arranged in order, the middle position is the one with an equal number of values on both sides.
2. For n values, the middle position is at position (n + 1)/2.
3. Example: For n = 7, the middle position is (7 + 1)/2 = 4th. There are 3 values before and 3 values after the 4th value.
4. The 4th value divides the data into two equal halves.

Case 2: Even number of observations (n is even)

1. When n is even, there is no single middle value.
2. The two "middle-most" values are at positions n/2 and (n/2) + 1.
3. Example: For n = 8, the two middle values are the 4th and 5th. There are 3 values before the 4th and 3 values after the 5th.
4. The median is defined as the average of these two values, ensuring it lies centrally.

Why Arrange in Order First?

• The concept of "middle" has no meaning unless the data is ordered.
• Without ordering, there is no way to determine which value divides the data into equal halves.
• Ascending and descending order give the same median.

Different Scenarios for Finding the Median:

1. Odd Number of Observations

• The median is exactly one of the data values.
• Example: Data = {3, 5, 7, 9, 11}. n = 5 (odd). Median = 3rd value = 7.

2. Even Number of Observations

• The median is the average of the two middle values.
• Example: Data = {4, 6, 8, 10}. n = 4 (even). Median = (6 + 8)/2 = 7.
• Note: The median (7) is NOT in the original data.

3. Data with Repeated Values

• If values repeat, still arrange in order and count positions.
• Example: {2, 3, 3, 5, 5, 5, 8}. n = 7 (odd). Median = 4th value = 5.

4. Data with Outliers

• The median is unaffected by extreme values.
• Example: {1, 2, 3, 4, 100}. Mean = 22, but Median = 3.
• The median (3) better represents the typical value.

5. All Values Equal

• If all observations are the same, the median equals that value.
• Example: {5, 5, 5, 5}. Median = (5 + 5)/2 = 5.

6. Two Observations

• n = 2 (even). Median = average of both values.
• Example: {3, 9}. Median = (3 + 9)/2 = 6.

Applications of Median:

• Income and Salary Analysis: Median income is used instead of mean income because a few very high earners can skew the mean upward. The median gives a better picture of what a "typical" person earns.
• Real Estate: Median house prices are reported because a few extremely expensive properties would inflate the mean. The median shows what a typical buyer pays.
• Education: Median marks or scores are used when the distribution is skewed. If most students score between 50–70 but a few score 100, the median represents the class better than the mean.
• Quality Control: In manufacturing, the median of product measurements is used to identify the typical product dimension, ignoring defective outliers.
• Medical Data: Median survival times are reported in medical studies because a few patients who live much longer (or shorter) would distort the mean.
• Sports Statistics: Median scores, times, or distances give a better sense of typical performance than averages affected by exceptional performances.