Question

Find the standard deviation ofthe average temperatures recorded over a five-day period last winter: 19, 21, 18, 24, 12?

Answer 1

 

Given Data:

\[ X = \{19, 21, 18, 24, 12\} \]

Formula for Sample Standard Deviation:

\[ s = \sqrt{\frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n-1}} \]

Step by Step Solution:

1. Find the Mean (\(\bar{X}\)): \[ \bar{X} = \frac{19 + 21 + 18 + 24 + 12}{5} = \frac{94}{5} = 18.8 \]
2. Subtract the Mean and Square the Result for Each Data Point: \[ (X_i - \bar{X})^2 \] \[ (19 - 18.8)^2, \ (21 - 18.8)^2, \ (18 - 18.8)^2, \ (24 - 18.8)^2, \ (12 - 18.8)^2 \]
3. Sum the Squared Differences: \[ \sum_{i=1}^{5} (X_i - \bar{X})^2 = (0.2)^2 + (2.2)^2 + (-0.8)^2 + (5.2)^2 + (-6.8)^2 = 78.8 \]
4. Divide by \(n-1\) (Number of Data Points Minus One): \[ \frac{\sum_{i=1}^{5} (X_i - \bar{X})^2}{n-1} = \frac{78.8}{5-1} = \frac{78.8}{4} = 19.7 \]
5. Take the Square Root: \[ s = \sqrt{19.7} \]

Answer:

\[ s \approx 4.44 \]