Question

calculate the standard deviation and variance of the sample quantitative data shown, to two decimal places.

x
10.5
7.1
27.6
14.9
18.1
14.4

standard deviation:
variance:
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question 4
the quantitative data was gathered by taking a random sample. calculate the standard deviation. round to one decimal place.

x
4
24
16
14
19

Explanation:

First Problem (Sample Data: 10.5, 7.1, 27.6, 14.9, 18.1, 14.4)

Step1: Calculate the sample mean ($\bar{x}$)

The formula for the sample mean is $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$, where $n = 6$ (number of data points) and $x_i$ are the data values.
$\sum x_i = 10.5 + 7.1 + 27.6 + 14.9 + 18.1 + 14.4 = 92.6$
$\bar{x} = \frac{92.6}{6} \approx 15.4333$

Step2: Calculate the squared deviations from the mean

For each data point $x_i$, calculate $(x_i - \bar{x})^2$:

• $(10.5 - 15.4333)^2 \approx (-4.9333)^2 \approx 24.337$
• $(7.1 - 15.4333)^2 \approx (-8.3333)^2 \approx 69.444$
• $(27.6 - 15.4333)^2 \approx (12.1667)^2 \approx 147.328$
• $(14.9 - 15.4333)^2 \approx (-0.5333)^2 \approx 0.284$
• $(18.1 - 15.4333)^2 \approx (2.6667)^2 \approx 7.111$
• $(14.4 - 15.4333)^2 \approx (-1.0333)^2 \approx 1.068$

Step3: Calculate the sum of squared deviations ($\sum (x_i - \bar{x})^2$)

$\sum (x_i - \bar{x})^2 \approx 24.337 + 69.444 + 147.328 + 0.284 + 7.111 + 1.068 = 249.572$

Step4: Calculate the sample variance ($s^2$)

The formula for sample variance is $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$. Here, $n - 1 = 5$.
$s^2 = \frac{249.572}{5} \approx 49.9144$

Step5: Calculate the sample standard deviation ($s$)

The standard deviation is the square root of the variance: $s = \sqrt{s^2}$
$s = \sqrt{49.9144} \approx 7.06$ (rounded to two decimal places)
The variance, rounded to two decimal places, is $49.91$.

Step1: Calculate the sample mean ($\bar{x}$)

$n = 5$ (number of data points). $\sum x_i = 4 + 24 + 16 + 14 + 19 = 77$
$\bar{x} = \frac{77}{5} = 15.4$

Step2: Calculate the squared deviations from the mean

• $(4 - 15.4)^2 = (-11.4)^2 = 129.96$
• $(24 - 15.4)^2 = (8.6)^2 = 73.96$
• $(16 - 15.4)^2 = (0.6)^2 = 0.36$
• $(14 - 15.4)^2 = (-1.4)^2 = 1.96$
• $(19 - 15.4)^2 = (3.6)^2 = 12.96$

Step3: Calculate the sum of squared deviations ($\sum (x_i - \bar{x})^2$)

$\sum (x_i - \bar{x})^2 = 129.96 + 73.96 + 0.36 + 1.96 + 12.96 = 219.2$

Step4: Calculate the sample variance ($s^2$)

Using $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$, with $n - 1 = 4$:
$s^2 = \frac{219.2}{4} = 54.8$

Step5: Calculate the sample standard deviation ($s$)

$s = \sqrt{54.8} \approx 7.4$ (rounded to one decimal place)

Answer:

Standard deviation: $7.06$
Variance: $49.91$

Second Problem (Sample Data: 4, 24, 16, 14, 19)