Question

this data is from a sample. calculate the mean, standard deviation, and variance.
data | 35.2 27.8 16.9 33.5 41.9 29.0 26.3 26.1 12.3 46.9
make sure to round appropriately
sample mean =
sample standard deviation =
sample variance =
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: $35.2 + 27.8+16.9 + 33.5+41.9+29.0+26.3+26.1+12.3+46.9$.
Calculating the sum: $35.2+27.8 = 63$; $63+16.9 = 79.9$; $79.9+33.5 = 113.4$; $113.4+41.9 = 155.3$; $155.3+29.0 = 184.3$; $184.3+26.3 = 210.6$; $210.6+26.1 = 236.7$; $236.7+12.3 = 249$; $249+46.9 = 295.9$.
There are $n = 10$ data points. So the mean $\bar{x}=\frac{295.9}{10}=29.59$.

Step2: Calculate the sample variance

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

• For $x = 35.2$: $(35.2 - 29.59)^{2}=(5.61)^{2}=31.4721$
• For $x = 27.8$: $(27.8 - 29.59)^{2}=(- 1.79)^{2}=3.2041$
• For $x = 16.9$: $(16.9 - 29.59)^{2}=(-12.69)^{2}=161.0361$
• For $x = 33.5$: $(33.5 - 29.59)^{2}=(3.91)^{2}=15.2881$
• For $x = 41.9$: $(41.9 - 29.59)^{2}=(12.31)^{2}=151.5361$
• For $x = 29.0$: $(29.0 - 29.59)^{2}=(-0.59)^{2}=0.3481$
• For $x = 26.3$: $(26.3 - 29.59)^{2}=(-3.29)^{2}=10.8241$
• For $x = 26.1$: $(26.1 - 29.59)^{2}=(-3.49)^{2}=12.1801$
• For $x = 12.3$: $(12.3 - 29.59)^{2}=(-17.29)^{2}=298.9441$
• For $x = 46.9$: $(46.9 - 29.59)^{2}=(17.31)^{2}=299.6361$

Now sum these squared differences: $31.4721+3.2041 = 34.6762$; $34.6762+161.0361 = 195.7123$; $195.7123+15.2881 = 211.0004$; $211.0004+151.5361 = 362.5365$; $362.5365+0.3481 = 362.8846$; $362.8846+10.8241 = 373.7087$; $373.7087+12.1801 = 385.8888$; $385.8888+298.9441 = 684.8329$; $684.8329+299.6361 = 984.469$.

Then, $s^{2}=\frac{984.469}{10 - 1}=\frac{984.469}{9}\approx109.3854$.

Step3: Calculate the sample standard deviation

The sample standard deviation $s=\sqrt{s^{2}}=\sqrt{109.3854}\approx10.46$.

Step4: Calculate the population standard deviation

The formula for population variance $\sigma^{2}=\frac{\sum_{i = 1}^{N}(x_{i}-\mu)^{2}}{N}$ (where $N$ is the population size and $\mu$ is the population mean, which is the same as the sample mean here since we use the same data).
We already have $\sum_{i = 1}^{N}(x_{i}-\mu)^{2}=984.469$ and $N = 10$. So $\sigma^{2}=\frac{984.469}{10}=98.4469$.
The population standard deviation $\sigma=\sqrt{98.4469}\approx9.92$.

Answer:

Sample Mean = $29.59$
Sample Standard Deviation = $10.46$ (rounded to two decimal places)
Sample Variance = $109.39$ (rounded to two decimal places)
Population Standard Deviation = $9.92$ (rounded to two decimal places)