To find the mean, median, and spread for the dataset represented by the bar graph, we first need to extract the values indicated by the heights of the bars:
- For \(10\) (height = 2): count = 2
- For \(11\) (height = 4): count = 4
- For \(12\) (height = 4): count = 4
- For \(13\) (height = 2): count = 2
This gives us the dataset:
- Two 10s: 10, 10
- Four 11s: 11, 11, 11, 11
- Four 12s: 12, 12, 12, 12
- Two 13s: 13, 13
Now we can represent the complete dataset as follows: \[ 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13 \]
Mean Calculation: To find the mean, we calculate the total sum of the dataset and divide it by the number of elements.
\[ \text{Sum} = 10 + 10 + 11 + 11 + 11 + 11 + 12 + 12 + 12 + 12 + 13 + 13 = 10 \times 2 + 11 \times 4 + 12 \times 4 + 13 \times 2 \] \[ = 20 + 44 + 48 + 26 = 138 \]
The number of elements is \( 12 \). Therefore, the mean is:
\[ \text{Mean} = \frac{138}{12} = 11.5 \]
Median Calculation: To find the median, we arrange the numbers in order (which they already are), and since we have an even number of elements (12), the median will be the average of the 6th and 7th values:
Ordered dataset: \[ 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13 \]
The 6th and 7th values are both 11 and 12 respectively.
\[ \text{Median} = \frac{11 + 12}{2} = \frac{23}{2} = 11.5 \]
Spread Calculation: The spread (range) is calculated as the difference between the maximum and minimum values in the dataset:
The maximum value is 13 and the minimum value is 10:
\[ \text{Spread} = 13 - 10 = 3 \]
So, the final values are:
- The mean = 11.5
- The median = 11.5
- The spread = 3
Thus:
- The mean = 11.5.
- The median = 11.5.
- The spread = 3.