What is the median of the following data? 2, 3, -1, 2, 6, 8, 9
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Finding the Median of a Data Set
The question asks us to find the median of the given set of data: 2, 3, -1, 2, 6, 8, 9.
Understanding the Median
The median is a measure of central tendency. It represents the middle value in a data set that has been arranged in numerical order. Unlike the mean (average), the median is not affected by extreme values (outliers), making it a useful measure for skewed distributions.
Steps to Calculate the Median
To find the median of a data set, follow these steps:
- Arrange the data points in ascending order (from smallest to largest).
- Count the total number of data points (let's call this 'n').
- Determine the median based on whether 'n' is odd or even:
- If 'n' is odd, the median is the middle value. The position of the median is the \(\left(\frac{n+1}{2}\right)\)-th term.
- If 'n' is even, the median is the average (mean) of the two middle values. The positions of the two middle values are the \(\left(\frac{n}{2}\right)\)-th term and the \(\left(\frac{n}{2} + 1\right)\)-th term.
Applying the Steps to the Given Data
Let's apply these steps to the data set: 2, 3, -1, 2, 6, 8, 9.
Step 1: Arrange the data in ascending order.
The ordered data set is: -1, 2, 2, 3, 6, 8, 9.
Step 2: Count the number of data points.
There are 7 data points in the set. So, \(n = 7\).
Step 3: Determine the median.
Since \(n = 7\) is an odd number, the median is the middle value. The position of the median is the \(\left(\frac{n+1}{2}\right)\)-th term.
Position of median = \(\left(\frac{7+1}{2}\right)\)-th term = \(\left(\frac{8}{2}\right)\)-th term = 4th term.
Now, let's find the 4th term in the ordered data set (-1, 2, 2, 3, 6, 8, 9).
- 1st term: -1
- 2nd term: 2
- 3rd term: 2
- 4th term: 3
- 5th term: 6
- 6th term: 8
- 7th term: 9
The 4th term in the ordered list is 3.
Therefore, the median of the data set {2, 3, -1, 2, 6, 8, 9} is 3.
| Original Data | Ordered Data | Position |
|---|---|---|
| 2 | -1 | 1st |
| 3 | 2 | 2nd |
| -1 | 2 | 3rd |
| 2 | 3 | 4th (Median) |
| 6 | 6 | 5th |
| 8 | 8 | 6th |
| 9 | 9 | 7th |
The calculated median is 3.
Revision Table: Median Calculation Steps
| Step | Action | Result for Data {-1, 2, 2, 3, 6, 8, 9} |
|---|---|---|
| 1 | Order data | -1, 2, 2, 3, 6, 8, 9 |
| 2 | Count data points (n) | n = 7 |
| 3a | Check if n is odd/even | n is odd |
| 3b | Find median position (\(\frac{n+1}{2}\)) | \(\frac{7+1}{2} = 4\)-th term |
| 4 | Identify the value at the median position | 3 (the 4th term) |
Additional Information: Measures of Central Tendency
The median is one of several measures used to describe the center of a data set. Other common measures include the mean and the mode.
- Mean: The average of all the data points. Calculated by summing all values and dividing by the number of values. Formula: \(\text{Mean} = \frac{\sum x}{n}\).
- Mode: The value that appears most frequently in the data set. A data set can have one mode (unimodal), multiple modes (multimodal), or no mode if all values appear only once.
For the given data set {-1, 2, 2, 3, 6, 8, 9}:
- Mean: \(\frac{-1 + 2 + 2 + 3 + 6 + 8 + 9}{7} = \frac{29}{7} \approx 4.14\).
- Median: 3 (as calculated above).
- Mode: 2 (appears twice, which is more than any other value).
Choosing the appropriate measure of central tendency depends on the nature of the data and the presence of outliers or skewed distributions. The median is often preferred when the data includes extreme values.
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