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:

$$s=\sqrt{\frac{\sum _{i=1}^n(X_i-\overline{X}{)}^2}{n-1}}$$

Step by Step Solution:

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

Answer:

$$s\approx 4.44$$