Finding the Median of Observations

The question asks us to find the median of a given set of observations: 10, 11, 9, 12, 10, 10, 12, 10, 9, 11.

Understanding the Median

The median is a measure of central tendency. It is the middle value in a data set that is arranged in ascending or descending order. The median divides the data set into two equal halves.

Steps to Calculate the Median

To find the median of a set of observations, follow these steps:

1. Arrange the observations in ascending order (from smallest to largest).
2. Count the total number of observations, denoted by 'n'.
3. Determine if 'n' is an odd or even number.
4. If 'n' is odd, the median is the observation at the \(\left(\frac{n+1}{2}\right)^{\text{th}}\) position.
5. If 'n' is even, the median is the average of the observations at the \(\left(\frac{n}{2}\right)^{\text{th}}\) position and the \(\left(\frac{n}{2} + 1\right)^{\text{th}}\) position.

Applying the Steps to the Given Observations

Let's apply these steps to the observations: 10, 11, 9, 12, 10, 10, 12, 10, 9, 11.

Step 1: Arrange in Ascending Order

Arranging the observations in ascending order, we get:

9, 9, 10, 10, 10, 10, 11, 11, 12, 12

Step 2: Count the Number of Observations (n)

Let's count the number of observations in the set:

9, 9, 10, 10, 10, 10, 11, 11, 12, 12

There are 10 observations. So, n = 10.

Step 3: Determine if n is Odd or Even

Since n = 10, which is an even number.

Step 4: Calculate the Median for Even n

For an even number of observations, the median is the average of the observations at the \(\left(\frac{n}{2}\right)^{\text{th}}\) position and the \(\left(\frac{n}{2} + 1\right)^{\text{th}}\) position.

• The \(\left(\frac{n}{2}\right)^{\text{th}}\) position is the \(\left(\frac{10}{2}\right)^{\text{th}} = 5^{\text{th}}\) position.
• The \(\left(\frac{n}{2} + 1\right)^{\text{th}}\) position is the \(\left(\frac{10}{2} + 1\right)^{\text{th}} = (5 + 1)^{\text{th}} = 6^{\text{th}}\) position.

Looking at the sorted observations (9, 9, 10, 10, 10, 10, 11, 11, 12, 12):

• The 5^th observation is 10.
• The 6^th observation is 10.

Now, we calculate the average of the 5^th and 6^th observations:

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

Median = \(\frac{10 + 10}{2}\)

Median = \(\frac{20}{2}\)

Median = 10

Conclusion

The median of the given observations 10, 11, 9, 12, 10, 10, 12, 10, 9, 11 is 10.

Revision Table: Median Calculation Steps

Additional Information: Measures of Central Tendency

The median is one of the main measures of central tendency, which are used to describe the center point of a data set. Other common measures include:

• Mean: The average of all observations. Calculated by summing all observations and dividing by the total number of observations.
• Mode: The observation that appears most frequently in the data set. A data set can have one mode (unimodal), more than one mode (multimodal), or no mode. In the given data (9, 9, 10, 10, 10, 10, 11, 11, 12, 12), the observation 10 appears 4 times, which is more frequent than any other observation. So, the mode is 10.

The choice of which measure of central tendency to use depends on the nature of the data and the presence of outliers. The median is often preferred over the mean for skewed data or data with outliers, as it is less affected by extreme values.