Question

for the following set of data, find the population standard deviation, to the nearest thousandth. 129, 114, 105, 84, 126, 124, 148 copy values for calculator open statistics calculator

Explanation:

Step1: Find the population mean ($\mu$)

The formula for the population mean is $\mu = \frac{\sum_{i = 1}^{n} x_i}{n}$, where $x_i$ are the data points and $n$ is the number of data points.
The data set is $129, 114, 105, 84, 126, 124, 148$. The number of data points $n = 7$.
First, calculate the sum of the data points:
$\sum_{i = 1}^{7} x_i=129 + 114+105 + 84+126+124+148$
$=129+114 = 243$; $243+105 = 348$; $348+84 = 432$; $432+126 = 558$; $558+124 = 682$; $682+148 = 830$
Then, $\mu=\frac{830}{7}\approx118.5714$

Step2: Calculate the squared differences from the mean

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

• For $x_1 = 129$: $(129 - 118.5714)^2=(10.4286)^2\approx108.7551$
• For $x_2 = 114$: $(114 - 118.5714)^2=(- 4.5714)^2\approx20.8984$
• For $x_3 = 105$: $(105 - 118.5714)^2=(-13.5714)^2\approx184.1837$
• For $x_4 = 84$: $(84 - 118.5714)^2=(-34.5714)^2\approx1195.1837$
• For $x_5 = 126$: $(126 - 118.5714)^2=(7.4286)^2\approx55.1837$
• For $x_6 = 124$: $(124 - 118.5714)^2=(5.4286)^2\approx29.4694$
• For $x_7 = 148$: $(148 - 118.5714)^2=(29.4286)^2\approx866.0408$

Step3: Find the sum of the squared differences

$\sum_{i = 1}^{7}(x_i-\mu)^2=108.7551+20.8984 + 184.1837+1195.1837+55.1837+29.4694+866.0408$
$=108.7551+20.8984=129.6535$; $129.6535 + 184.1837 = 313.8372$; $313.8372+1195.1837 = 1509.0209$; $1509.0209+55.1837 = 1564.2046$; $1564.2046+29.4694 = 1593.674$; $1593.674+866.0408 = 2459.7148$

Step4: Calculate the population variance ($\sigma^2$)

The formula for population variance is $\sigma^2=\frac{\sum_{i = 1}^{n}(x_i - \mu)^2}{n}$
We have $\sum_{i = 1}^{n}(x_i - \mu)^2 = 2459.7148$ and $n = 7$
So, $\sigma^2=\frac{2459.7148}{7}\approx351.3878$

Step5: Calculate the population standard deviation ($\sigma$)

The population standard deviation is the square root of the variance, so $\sigma=\sqrt{\sigma^2}$
$\sigma=\sqrt{351.3878}\approx18.745$

Answer:

$18.745$