MEDIAN example and Calculation with Sample MCQs Summary statistics | step-by-step examples

 

MEDIAN

The median is a measure of central tendency representing a dataset's middle value when arranged in order. Here are some examples of the advantages and disadvantages of using the median for both grouped and ungrouped data:

Advantages of using median for ungrouped data:

• Not affected by extreme values: The median is a robust measure of central tendency that is not affected by extreme values or outliers in the dataset.
• Easy to calculate: The median is easy to calculate and does not require complicated formulas or calculations.
• Useful for skewed distributions: The median is particularly useful for datasets with skewed distributions, where the mean may not accurately represent the central tendency.

Disadvantages of using median for ungrouped data:

• May not represent the entire dataset: The median only represents the middle value in the dataset and does not provide information about the distribution of the data.
• Limited inferences: The median provides limited information about the variability of the data, making it difficult to draw conclusions about the spread of the dataset.

Advantages of using median for grouped data:

• Useful for large datasets: The median is particularly useful for large datasets where calculating the mean would be impractical.
• Provides information about the distribution: The median can provide information about the distribution of the data when presented in a grouped frequency distribution table.

Disadvantages of using median for grouped data:

• Can be less precise: When working with grouped data, the median can be less precise than when working with ungrouped data.

• May be influenced by grouping intervals: The median can be influenced by the grouping intervals used in a frequency distribution table, which can lead to inaccurate results if the intervals are not chosen carefully.

MEDIAN 

EXAMPLE QUESTIONS

Ungrouped Data:

The formula for finding the median for a set of ungrouped data depends on whether the number of data points is odd or even.

Odd Number of Data Points:

If the number of data points is odd, the median is simply the middle value when the data are arranged in order. For example, if we have the data set: 2, 5, 7, 8, 10, 11, 12, the median is 8, which is the fourth value in the ordered list.

Even Number of Data Points:

If the number of data points is even, the median is calculated as the average of the two middle values when the data are arranged in order. For example, if we have the data set: 2, 5, 7, 8, 10, 11, the median is the average of the two middle values: (7 + 8) / 2 = 7.5.

In both cases, the formula for finding the median is simply a matter of finding the middle value(s) of the ordered data set.

Question: Find the median of the following set of numbers: 2, 5, 7, 8, 10, 11, 12.

Solution:

To find the median, we need to arrange the numbers in order:

2, 5, 7, 8, 10, 11, 12

Since there are an odd number of numbers, the median is the middle value of 8.

Question: Given the following set of data: 1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 9, 9, 10, 10, 10. Find the median.

Solution:

To find the median, we need to arrange the numbers in order:

1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 9, 9, 10, 10, 10

Since there are an odd number of numbers, the median is the middle value, 7.

Grouped Data:

When working with grouped data, the median is calculated by finding the interval containing the median value and then using a formula to estimate the exact median value within that interval.

The formula for finding the median for grouped data is:

Median = \(L+\frac{h}{f}(\frac {\sum f}{2}-C.F)\)

Where:

L = lower limit of the interval containing the median

\(\sum f\) = total frequency

C.F = cumulative frequency up to the interval containing the median

f = frequency of the median interval

h = interval size

To use this formula, we need to first determine which interval contains the median value. To do this, we calculate the cumulative frequency (CF) for each interval, which is the sum of the frequencies of all the intervals up to and including the current interval.

Once we have identified the interval containing the median value, we can use the formula above to estimate the exact value of the median within that interval. The formula works by finding the distance between the median and the lower limit of the interval (\(\sum f\)/2 - CF), dividing this by the frequency of the median interval (f), and multiplying the result by the interval size (i). Finally, we add this result to the lower limit of the median interval (L) to get the estimated value of the median.

Question: Find the median for the following frequency distribution table:

Class Interval Frequency 0-10 5 10-20 8 20-30 10 30-40 7 40-50 4

Solution: To find the median, we first need to calculate the cumulative frequency (CF) for each interval:

Class Interval Frequency CF 0-10 5 5 10-20 8 13 20-30 10 23 30-40 7 30 40-50 4 34

                Median = \(L+\frac{h}{f}(\frac {\sum f}{2}-C.F)\)

Where: L = lower limit of the interval containing the median n = total frequency CF = cumulative frequency up to the interval containing the median f = frequency of the median interval h = interval size We can calculate the median as follows: 

 Median = 20 + (((17 - 13) / 8) x 10) = 22.5 

 Therefore, the median of the data is 22.5.

Ungrouped Data:

• Find the median of the following set of numbers: 2, 5, 7, 8, 10, 11, 12.
• Given the following set of data: 1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 9, 9, 10, 10, 10. Find the median.
• If the median of a set of 9 numbers is 7, what is the value of the 5th number?

Grouped Data:

• Find the median for the following frequency distribution table:

Class Interval Frequency 0-10 5 10-20 8 20-30 10 30-40 7 40-50 4