Question

this data is from a sample. calculate the mean, standard deviation, and variance.
x 34.4 10.8 34.7 43.5 13.6 11.8 15.4
please show the following answers to 2 decimal places.
sample mean =
sample standard deviation =
sample variance =
oops - now you discover that the data was actually from a population! so now you must give the population standard deviation.
population standard deviation =

Explanation:

Step1: Calculate the sample mean

The formula for the sample mean $\bar{x}$ is $\bar{x} = \frac{\sum_{i = 1}^{n} x_i}{n}$, where $x_i$ are the data points and $n$ is the number of data points.
First, sum the data points: $34.4 + 10.8 + 34.7 + 43.5 + 13.6 + 11.8 + 15.4 = 34.4+10.8 = 45.2; 45.2+34.7 = 79.9; 79.9+43.5 = 123.4; 123.4+13.6 = 137; 137+11.8 = 148.8; 148.8+15.4 = 164.2$.
There are $n = 7$ data points. So the mean is $\frac{164.2}{7} \approx 23.46$.

Step2: Calculate the sample variance

The formula for sample variance $s^2$ is $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}$.
First, calculate each $(x_i-\bar{x})^2$:

• For $x_1 = 34.4$: $(34.4 - 23.46)^2=(10.94)^2 = 119.6836$
• For $x_2 = 10.8$: $(10.8 - 23.46)^2=(-12.66)^2 = 160.2756$
• For $x_3 = 34.7$: $(34.7 - 23.46)^2=(11.24)^2 = 126.3376$
• For $x_4 = 43.5$: $(43.5 - 23.46)^2=(20.04)^2 = 401.6016$
• For $x_5 = 13.6$: $(13.6 - 23.46)^2=(-9.86)^2 = 97.2196$
• For $x_6 = 11.8$: $(11.8 - 23.46)^2=(-11.66)^2 = 135.9556$
• For $x_7 = 15.4$: $(15.4 - 23.46)^2=(-8.06)^2 = 64.9636$

Sum these squared differences: $119.6836+160.2756 = 279.9592; 279.9592+126.3376 = 406.2968; 406.2968+401.6016 = 807.8984; 807.8984+97.2196 = 905.118; 905.118+135.9556 = 1041.0736; 1041.0736+64.9636 = 1106.0372$.
Then, divide by $n - 1=6$: $s^2=\frac{1106.0372}{6}\approx184.34$.

Step3: Calculate the sample standard deviation

The sample standard deviation $s$ is the square root of the sample variance: $s=\sqrt{184.34}\approx13.58$.

Step4: Calculate the population variance and standard deviation

For a population, the variance $\sigma^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}$. We already have the sum of squared differences as $1106.0372$, and $n = 7$. So $\sigma^2=\frac{1106.0372}{7}\approx158.0053$.
The population standard deviation $\sigma$ is the square root of the population variance: $\sigma=\sqrt{158.0053}\approx12.57$.

Answer:

Sample Mean = $23.46$
Sample Standard Deviation = $13.58$
Sample Variance = $184.34$
Population Standard Deviation = $12.57$