If mean of the numbers 2, (2p + 2), 7, 13, 17, 4 and (p - 1) is 8, then find their median.
(c) 7
Understanding the Problem: Mean and Median
The question asks us to find the median of a set of numbers after first determining the value of an unknown variable, 'p', using the given mean. The set of numbers is 2, (2p + 2), 7, 13, 17, 4, and (p - 1). We are given that the mean of these 7 numbers is 8.
To solve this problem, we need to follow these steps:
- Calculate the sum of all the given numbers, including the expressions with 'p'.
- Use the formula for the mean to set up an equation involving 'p'.
- Solve the equation to find the value of 'p'.
- Substitute the value of 'p' back into the expressions (2p + 2) and (p - 1) to get the numerical values.
- List all 7 numbers with their numerical values.
- Arrange the numbers in ascending order.
- Identify the median, which is the middle value in the sorted list of numbers.
Step-by-Step Solution: Finding the Value of p
The given numbers are: 2, \((2p + 2)\), 7, 13, 17, 4, \((p - 1)\).
There are 7 numbers in total.
The mean of these numbers is given as 8.
The formula for the mean is:
\(\text{Mean} = \frac{\text{Sum of all numbers}}{\text{Total number of numbers}}\)
First, let's find the sum of the numbers:
\(\text{Sum} = 2 + (2p + 2) + 7 + 13 + 17 + 4 + (p - 1)\)
Combine the terms with 'p' and the constant terms:
\(\text{Sum} = (2p + p) + (2 + 2 + 7 + 13 + 17 + 4 - 1)\)
\(\text{Sum} = 3p + (45 - 1)\)
\(\text{Sum} = 3p + 44\)
Now, substitute the sum and the given mean into the mean formula:
\(8 = \frac{3p + 44}{7}\)
To solve for 'p', multiply both sides of the equation by 7:
\(8 \times 7 = 3p + 44\)
\(56 = 3p + 44\)
Subtract 44 from both sides:
\(56 - 44 = 3p\)
\(12 = 3p\)
Divide both sides by 3:
\(p = \frac{12}{3}\)
\(p = 4\)
So, the value of 'p' is 4.
Determining the Actual Numbers
Now that we know \(p = 4\), we can find the numerical values of the expressions \((2p + 2)\) and \((p - 1)\):
- \(2p + 2 = 2(4) + 2 = 8 + 2 = 10\)
- \(p - 1 = 4 - 1 = 3\)
The complete set of numbers is now:
2, 10, 7, 13, 17, 4, 3.
Finding the Median of the Numbers
To find the median, we need to arrange the numbers in ascending order:
2, 3, 4, 7, 10, 13, 17
There are 7 numbers in this set. For an odd number of data points, the median is the middle value. The position of the median is given by the formula \(\frac{n+1}{2}\), where 'n' is the total number of data points.
Here, \(n = 7\). So the median is the \(\frac{7+1}{2} = \frac{8}{2} = 4^{th}\) number in the ordered list.
Looking at the ordered list (2, 3, 4, 7, 10, 13, 17), the \(4^{th}\) number is 7.
Therefore, the median of the numbers is 7.
| Original Numbers | Sum Calculation | Equation for Mean | Solving for p | Numbers after Finding p | Ordered Numbers | Median |
|---|---|---|---|---|---|---|
| 2, (2p+2), 7, 13, 17, 4, (p-1) | \(3p + 44\) | \(8 = \frac{3p + 44}{7}\) | \(p = 4\) | 2, 10, 7, 13, 17, 4, 3 | 2, 3, 4, 7, 10, 13, 17 | 7 |
Revision Table: Mean and Median Concepts
Let's quickly review the definitions of mean and median.
| Term | Definition | How to Calculate |
|---|---|---|
| Mean | The average of a set of numbers. | Sum of all values divided by the total number of values. |
| Median | The middle value in a dataset that is ordered from least to greatest. | Arrange data in order. If 'n' is odd, median is the \(\left(\frac{n+1}{2}\right)^{\text{th}}\) value. If 'n' is even, median is the average of the \(\left(\frac{n}{2}\right)^{\text{th}}\) and \(\left(\frac{n}{2}+1\right)^{\text{th}}\) values. |
Additional Information: Measures of Central Tendency
Mean and median are both measures of central tendency. They tell us about the center or typical value of a dataset. Another common measure is the mode, which is the value that appears most frequently in the dataset.
- Mean: Good for symmetrical data without outliers. Affected by extreme values.
- Median: Good for skewed data or data with outliers. Not affected by extreme values.
- Mode: Good for categorical data or finding the most popular item. A dataset can have no mode, one mode, or multiple modes.
Understanding mean and median is fundamental in statistics and data analysis. This problem combined algebra (solving for 'p') with statistics (calculating mean and median) to reinforce both concepts.
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