To calculate the standard deviation of the given data set \(16, 19, 23, 23, 24, 27, 27, 28\), we will follow these steps:
- Calculate the mean (average) of the data set.
- Calculate the variance by finding the average of the squared differences from the mean.
- Take the square root of the variance to find the standard deviation.
Step 1: Calculate the Mean
\[ \text{Mean} = \frac{\sum \text{data}}{N} \]
where \(N\) is the number of data points.
\[ \text{Mean} = \frac{16 + 19 + 23 + 23 + 24 + 27 + 27 + 28}{8} = \frac{ 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28}{8} = \frac{ 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28}{8} = \frac{ 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28}{8} = \frac{ 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28}{8} = \frac{ 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28}{8} = \frac{ 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28}{8} = \frac{ 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28}{8} \]
The sum is \(16 + 19 + 23 + 23 + 24 + 27 + 27 + 28 = 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28 = 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28 = 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28 = 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28 = 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28 = 16 + 19 + 23 + 23 + 24 + 27 + 27 + 28 = 192\).
So,
\[ \text{Mean} = \frac{192}{8} = 24 \]
Step 2: Calculate the Variance
The variance is calculated as follows:
\[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N} \]
Calculating each squared deviation:
- \((16 - 24)^2 = 64\)
- \((19 - 24)^2 = 25\)
- \((23 - 24)^2 = 1\)
- \((23 - 24)^2 = 1\)
- \((24 - 24)^2 = 0\)
- \((27 - 24)^2 = 9\)
- \((27 - 24)^2 = 9\)
- \((28 - 24)^2 = 16\)
Now, summing these squared deviations:
\[ 64 + 25 + 1 + 1 + 0 + 9 + 9 + 16 = 125 \]
Now divide by \(N\) (the number of data points, which is 8):
\[ \text{Variance} = \frac{125}{8} = 15.625 \]
Step 3: Calculate Standard Deviation
Finally, take the square root of the variance:
\[ \text{Standard Deviation} = \sqrt{15.625} \approx 3.9 \]
Conclusion
Therefore, the standard deviation of the data set rounded to the nearest tenth is 3.9.