Let's go through each statistical measure step-by-step:
### 1. Mean (Average)
To find the mean, sum up all the values and then divide by the number of values.
[tex]\[ \text{Sum} = 7 + 7 + 7 + 14 + 14 + 12 + 4 + 14 + 13 + 1 = 93 \][/tex]
There are 10 values in total.
So, the mean is:
[tex]\[ \text{Mean} = \frac{\text{Sum}}{\text{Number of values}} = \frac{93}{10} = 9.3 \][/tex]
### 2. Median
To find the median, first sort the data in ascending order:
[tex]\[ 1, 4, 7, 7, 7, 12, 13, 14, 14, 14 \][/tex]
Since there is an even number of values (10), the median is the average of the 5th and 6th values in the sorted list.
The 5th value is 7, and the 6th value is 12.
So, the median is:
[tex]\[ \text{Median} = \frac{7 + 12}{2} = \frac{19}{2} = 9.5 \][/tex]
### 3. Mode
The mode is the value that appears most frequently. From the sorted list:
[tex]\[ 1, 4, 7, 7, 7, 12, 13, 14, 14, 14 \][/tex]
The numbers 7 and 14 both appear three times, so there are two modes.
Hence, the modes are:
[tex]\[ \text{Mode} = 7 \text{ and } 14 \][/tex]
### 4. Standard Deviation
To find the standard deviation, first compute the variance, which is the average of the squared differences from the mean.
The formula for variance (σ²) is:
[tex]\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 \][/tex]
Where [tex]\( x_i \)[/tex] are the individual data points, [tex]\( \mu \)[/tex] is the mean, and [tex]\( n \)[/tex] is the number of data points.
First, calculate the squared differences from the mean:
[tex]\[ (7 - 9.3)^2 = 5.29 \][/tex]
[tex]\[ (7 - 9.3)^2 = 5.29 \][/tex]
[tex]\[ (7 - 9.3)^2 = 5.29 \][/tex]
[tex]\[ (14 - 9.3)^2 = 22.09 \][/tex]
[tex]\[ (14 - 9.3)^2 = 22.09 \][/tex]
[tex]\[ (12 - 9.3)^2 = 7.29 \][/tex]
[tex]\[ (4 - 9.3)^2 = 28.09 \][/tex]
[tex]\[ (14 - 9.3)^2 = 22.09 \][/tex]
[tex]\[ (13 - 9.3)^2 = 13.69 \][/tex]
[tex]\[ (1 - 9.3)^2 = 68.89 \][/tex]
Sum of squared differences:
[tex]\[ 5.29 + 5.29 + 5.29 + 22.09 + 22.09 + 7.29 + 28.09 + 22.09 + 13.69 + 68.89 = 200.1 \][/tex]
Now divide by the number of values (10) to get the variance:
[tex]\[ \sigma^2 = \frac{200.1}{10} = 20.01 \][/tex]
The standard deviation (σ) is the square root of the variance:
[tex]\[ \sigma = \sqrt{20.01} \approx 4.47 \][/tex]
### Summary
- Mean: [tex]\(9.3\)[/tex]
- Median: [tex]\(9.5\)[/tex]
- Mode: [tex]\(7\)[/tex] and [tex]\(14\)[/tex] (bimodal)
- Standard Deviation: [tex]\(4.47\)[/tex]
