Question

we are going to learn how to calculate standard deviation by hand.
interestingly, in the real - world no statistician would ever calculate standard deviation by hand. the calculations involved are somewhat complex, and the risk of making a mistake is high. also, calculating by hand is slow. very slow. this is why statisticians rely on spreadsheets and computer programs to crunch their numbers.
so, why are we taking time to learn a process statisticians dont actually use? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works. this insight is valuable instead of viewing standard deviation as some magical number a spreadsheet or computer program gives us, we will be able to explain where that number comes from. the formula for standard deviation (sd) is $sd = sqrt{\frac{sum(x - mu)^2}{n}}$ where $sum$ means \sum of\, $x$ is a value in the data set, $mu$ is the mean of the data set, and $n$ is the number of data points in the population.
the standard - deviation formula may look confusing, but it will make sense after we break it down. guided steps:
step 1: find the mean.
step 2: for each data point, find the square of its distance to the mean.
step 3: sum the values from step 2.
step 4: divide by the number of data points.
step 5: take the square root.
here is our data set: 6, 3, 3, 1
step 1: finding $mu$ in $sqrt{\frac{sum(x - mu)^2}{n}}$
in this step, we find the mean of the data set, which is represented by the variable $mu$.
fill in the blank with your answer.
step 2: finding $(x - mu)^2$ in $sqrt{\frac{sum(x - mu)^2}{n}}$
in this step, we find the distance from each data point to the mean (i.e., the deviation) and square each of those distances.
for example, the first data point is 6 and the mean is 3, so the distance between them is 3. squaring this distance gives us 9.
complete the table below.
data point $x$ square of the distance from the mean $(x - mu)^2$
6 9
3
3
1
put each answer into a different blank.
step 3: finding $sum(x - mu)^2$ in $sqrt{\frac{sum(x - mu)^2}{n}}$
the symbol $sum$ means \sum\, so in this step we add up the four values we found in step 2.
fill in the blank.
$sum(x - mu)^2=$
step 4: finding $\frac{sum(x - mu)^2}{n}$ in $sqrt{\frac{sum(x - mu)^2}{n}}$
in this step, we divide our result from step 3 by the variable $n$, which is the number of data points.
fill in the blank.
$\frac{sum(x - mu)^2}{n}=$
step 5: finding the standard deviation $sqrt{\frac{sum(x - mu)^2}{n}}$
were almost finished! just take the square root of the answer from step 4 and were done.
fill in the blank.
round your answer to the nearest hundredth.
$sd=sqrt{\frac{sum(x - mu)^2}{n}}=$
lets practice math

Explanation:

Answer:

(Step 1) \( \mu = 3 \)
(Step 2) For \(x = 6\), \((x - \mu)^2=(6 - 3)^2 = 9\); for \(x = 3\), \((x - \mu)^2=(3 - 3)^2 = 0\); for \(x = 3\), \((x - \mu)^2=(3 - 3)^2 = 0\); for \(x = 1\), \((x - \mu)^2=(1 - 3)^2 = 4\)
(Step 3) \( \sum(x-\mu)^2=9 + 0+0 + 4=13\)
(Step 4) \(\frac{\sum(x - \mu)^2}{N}=\frac{13}{4}=3.25\)
(Step 5) \(SD=\sqrt{3.25}\approx1.80\)