Calculating the Median of a Data Set

The question asks us to find the median of the given set of numbers: 2, 9, 11, 7, 3, 4, 8, 5, 3, 6.

The median is a measure of central tendency that represents the middle value in a data set that has been ordered from least to greatest. To find the median, we follow these steps:

1. Arrange the data set in ascending order (from smallest to largest).
2. Count the total number of observations in the data set. Let this number be 'n'.
3. Determine if 'n' is an odd or an even number.
4. If 'n' is odd, the median is the value at the position \(\frac{n+1}{2}\) in the ordered data set.
5. If 'n' is even, the median is the average of the values at the positions \(\frac{n}{2}\) and \((\frac{n}{2} + 1)\) in the ordered data set.

Step-by-Step Median Calculation

Let's apply these steps to the given numbers: 2, 9, 11, 7, 3, 4, 8, 5, 3, 6.

Step 1: Arrange the data in ascending order.

The sorted data set is: 2, 3, 3, 4, 5, 6, 7, 8, 9, 11.

Step 2: Count the number of observations (n).

There are 10 numbers in the data set. So, n = 10.

Step 3: Determine if n is odd or even.

Since n = 10, the number of observations is even.

Step 4: Find the median for an even number of observations.

For an even data set, the median is the average of the two middle terms. These terms are located at positions \(\frac{n}{2}\) and \((\frac{n}{2} + 1)\).

• The position of the first middle term is \(\frac{n}{2} = \frac{10}{2} = 5\). This is the 5th term.
• The position of the second middle term is \((\frac{n}{2} + 1) = (5 + 1) = 6\). This is the 6th term.

Now, let's find the values of the 5th and 6th terms in our sorted data set (2, 3, 3, 4, 5, 6, 7, 8, 9, 11):

• 1st term: 2
• 2nd term: 3
• 3rd term: 3
• 4th term: 4
• 5th term: 5
• 6th term: 6
• 7th term: 7
• 8th term: 8
• 9th term: 9
• 10th term: 11

The 5th term is 5, and the 6th term is 6.

To find the median, we take the average of these two terms:

Median = \(\frac{\text{5th term} + \text{6th term}}{2}\)

Median = \(\frac{5 + 6}{2}\)

Median = \(\frac{11}{2}\)

Median = 5.5

Thus, the median of the given data set is 5.5.

Summary of Data and Median Calculation

Revision Table: Key Concepts for Median

Additional Information on Measures of Central Tendency

The median is one of the main measures of central tendency. Other important measures include the mean and the mode.

• Mean (Average): Calculated by summing all the values in a data set and dividing by the total number of values. It is affected by extreme values (outliers).
• Mode: The value that appears most frequently in a data set. A data set can have one mode ( unimodal), multiple modes (multimodal), or no mode if all values appear with the same frequency.
• Median: As discussed, the middle value of an ordered data set. It is less affected by extreme values compared to the mean, making it a better measure of central tendency for skewed distributions.

Choosing the appropriate measure of central tendency depends on the nature of the data and the goal of the analysis. For this question, we focused specifically on finding the median.