The median of the numbers 3, 1, 1, 5, 2 and 7 is:
2.5
Finding the Median of a Set of Numbers
The question asks us to find the median of the given set of numbers: 3, 1, 1, 5, 2, and 7. The median is a measure of central tendency. It represents the middle value in a dataset that has been ordered from least to greatest.
Steps to Calculate the Median
To find the median of a set of numbers, we follow these steps:
- Arrange the numbers in ascending order (from smallest to largest).
- Count the total number of data points.
- If the total number of data points is odd, the median is the middle number.
- If the total number of data points is even, the median is the average (mean) of the two middle numbers.
Applying the Steps to the Given Numbers
The given numbers are: 3, 1, 1, 5, 2, 7.
Step 1: Arrange the numbers in ascending order.
Let's arrange the numbers from smallest to largest:
1, 1, 2, 3, 5, 7
Step 2: Count the total number of data points.
There are 6 numbers in the set.
Step 3 & 4: Determine if the count is odd or even and find the median.
The total number of data points is 6, which is an even number. When the number of data points is even, the median is the average of the two middle numbers.
In the ordered list (1, 1, 2, 3, 5, 7), the middle two numbers are the 3rd and 4th numbers.
- The 3rd number is 2.
- The 4th number is 3.
To find the median, we calculate the average of 2 and 3:
Median = $\frac{\text{Sum of the two middle numbers}}{\text{Number of middle numbers}}$
Median = $\frac{2 + 3}{2}$
Median = $\frac{5}{2}$
Median = $2.5$
Therefore, the median of the numbers 3, 1, 1, 5, 2, and 7 is 2.5.
Let's review the options provided:
| Option | Value |
|---|---|
| 1 | 3.5 |
| 2 | 2 |
| 3 | 3 |
| 4 | 2.5 |
Our calculated median, 2.5, matches Option 4.
Revision Table: Key Concepts for Median
| Concept | Description | How to Find |
|---|---|---|
| Median | The middle value in an ordered dataset. It is a measure of central tendency that is not affected by extreme outliers. | 1. Order data. 2. Find middle value(s). 3. If odd count, median is the single middle value. 4. If even count, median is the average of the two middle values. |
| Ascending Order | Arranging numbers from the smallest value to the largest value. | Example: 1, 2, 3, 5, 7 |
| Average (Mean) | The sum of a set of numbers divided by the count of numbers in the set. Used to calculate the median for an even number of data points. | Formula: $\frac{\text{Sum of numbers}}{\text{Count of numbers}}$ |
Additional Information: Median vs. Mean vs. Mode
The median is one type of average. Other common measures of central tendency include the mean and the mode.
- Mean: The arithmetic average. Calculated by summing all the numbers and dividing by the count of numbers. For the set 1, 1, 2, 3, 5, 7, the mean is $\frac{1+1+2+3+5+7}{6} = \frac{19}{6} \approx 3.17$. The mean can be affected by outliers.
- Median: The middle value when the data is ordered. For the set 1, 1, 2, 3, 5, 7, the median is 2.5. The median is less affected by outliers compared to the mean.
- Mode: The value that appears most frequently in the dataset. For the set 1, 1, 2, 3, 5, 7, the number 1 appears twice, which is more than any other number. So, the mode is 1. There can be one mode (unimodal), multiple modes (multimodal), or no mode (if all values appear with the same frequency).
Understanding these different measures helps in analyzing data distribution.
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Important Questions from Elementary Statistics
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