Question

find and interpret the standard deviation of the data set.

1. 18, 11, 15, 20, 16
2. 78, 71, 68, 75, 46, 66

Explanation:

Problem 4:

Step 1: Find the mean (\(\mu\))

The data set is \( 18, 11, 15, 20, 16 \). The number of data points \( n = 5 \).

$$ \mu=\frac{18 + 11+15 + 20+16}{5}=\frac{80}{5} = 16 $$

Step 2: Find the squared differences from the mean

For each data point \( x \), calculate \( (x-\mu)^2 \):

• For \( x = 18 \): \( (18 - 16)^2=2^2 = 4 \)
• For \( x = 11 \): \( (11 - 16)^2=(- 5)^2=25 \)
• For \( x = 15 \): \( (15 - 16)^2=(-1)^2 = 1 \)
• For \( x = 20 \): \( (20 - 16)^2=4^2=16 \)
• For \( x = 16 \): \( (16 - 16)^2=0^2 = 0 \)

Step 3: Find the variance (\(\sigma^2\))

Variance is the average of the squared differences.

$$ \sigma^2=\frac{4 + 25+1+16 + 0}{5}=\frac{46}{5}=9.2 $$

Step 4: Find the standard deviation (\(\sigma\))

Standard deviation is the square root of the variance.

$$ \sigma=\sqrt{9.2}\approx3.24 $$

Problem 5:

Step 1: Find the mean (\(\mu\))

The data set is \( 78, 71, 68, 75, 46, 66 \). The number of data points \( n = 6 \).

$$ \mu=\frac{78+71 + 68+75+46+66}{6}=\frac{404}{6}\approx67.33 $$

Step 2: Find the squared differences from the mean

For each data point \( x \), calculate \( (x - \mu)^2 \):

• For \( x = 78 \): \( (78-67.33)^2=(10.67)^2\approx113.85 \)
• For \( x = 71 \): \( (71 - 67.33)^2=(3.67)^2\approx13.47 \)
• For \( x = 68 \): \( (68 - 67.33)^2=(0.67)^2\approx0.45 \)
• For \( x = 75 \): \( (75 - 67.33)^2=(7.67)^2\approx58.83 \)
• For \( x = 46 \): \( (46 - 67.33)^2=(-21.33)^2\approx455.07 \)
• For \( x = 66 \): \( (66 - 67.33)^2=(-1.33)^2\approx1.77 \)

Step 3: Find the variance (\(\sigma^2\))

$$ \sigma^2=\frac{113.85+13.47 + 0.45+58.83+455.07+1.77}{6}=\frac{643.44}{6} = 107.24 $$

Step 4: Find the standard deviation (\(\sigma\))

$$ \sigma=\sqrt{107.24}\approx9.72 $$

Answer:

The standard deviation is approximately \( 3.24 \). The data values are relatively close to the mean (average distance of about 3.24 from the mean).