The heights (in cm) of 5 students are 150, 165, 161, 144, and 155. What are the values of mean and median (in cm) respectively?
155 and 155
Understanding Mean and Median Calculations
The question asks us to find the mean and the median for a given set of heights of 5 students. The heights are 150 cm, 165 cm, 161 cm, 144 cm, and 155 cm.
Let's calculate each measure of central tendency step-by-step.
Calculating the Mean Height
The mean (or average) is calculated by summing all the values in a dataset and then dividing by the total number of values. The formula for the mean is:
\(\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}\)
In this case, the values are the heights of the 5 students:
- Sum of heights = \(150 + 165 + 161 + 144 + 155\)
- Sum of heights = \(775\) cm
- Number of students = \(5\)
Now, we calculate the mean:
\(\text{Mean} = \frac{775}{5}\)
\(\text{Mean} = 155\) cm
So, the mean height of the students is 155 cm.
Calculating the Median Height
The median is the middle value in a dataset that is arranged in ascending or descending order. When the number of observations (n) is odd, the median is the \(\left(\frac{n+1}{2}\right)^{\text{th}}\) observation after arranging the data.
First, let's arrange the given heights in ascending order:
144, 150, 155, 161, 165
The number of observations is n = 5, which is an odd number.
The position of the median is \(\left(\frac{5+1}{2}\right)^{\text{th}}\) = \(\left(\frac{6}{2}\right)^{\text{th}}\) = \(3^{\text{rd}}\) position.
Looking at the sorted list, the value at the 3rd position is 155.
Sorted heights: 144, 150, 155, 161, 165
So, the median height of the students is 155 cm.
Summary of Results
Based on our calculations:
- Mean height = 155 cm
- Median height = 155 cm
The values of mean and median respectively are 155 and 155.
| Statistic | Value (cm) |
|---|---|
| Mean | 155 |
| Median | 155 |
Revision Table: Key Statistics Concepts
Here is a quick look at the measures of central tendency discussed:
| Measure | Definition | Calculation Method |
|---|---|---|
| Mean | The average of all data points. | Sum of values divided by the count of values. |
| Median | The middle value in an ordered dataset. | Order data; find the middle value (or average of two middle values if count is even). |
| Mode | The value that appears most frequently in the dataset. | Identify the value with the highest frequency. (Not calculated in this problem) |
Additional Information on Calculating Median
The method for calculating the median depends on whether the number of data points (n) is odd or even.
- If n is odd: Arrange the data in order and the median is the value at the \(\left(\frac{n+1}{2}\right)^{\text{th}}\) position.
- If n is even: Arrange the data in order and the median is the average of the values at the \(\left(\frac{n}{2}\right)^{\text{th}}\) and \(\left(\frac{n}{2}+1\right)^{\text{th}}\) positions.
In this specific problem, n=5 (odd), so the median is the value at the \(\left(\frac{5+1}{2}\right)^{\text{th}}\) = 3rd position in the sorted list.
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