[Math] Standard deviation (percentage)

statistics

I'm writing an exam tomorrow and I just can't figure out how to do this sum.
I did the standard deviation already now all I need to know is how to determine the percentage of workers. Sounds weird but I will give you the question

Sample of weekly wages earned by $10$ workers

$2250, 2250, 3000, 3300, 3300, 3600, 3900, 4350, 5250$

Calculate the mean, which is $3555$.

Calculate the standard deviation, which is $900.12$.

Now this is where I can't figure out how to do it .

Determine the percentage of workers which lie within ONE standard deviation of the mean?

Please help me I don't know what formula to use for this question

Best Answer

This is an exercise illustrating the Empirical Rule:

To start, let's get the correct mean $\bar X = 3466.667$ and standard deviation $S = 961.7692.$ (Please do some checking here. Did one of us type the observations incorrectly, or did you incorrectly compute $\bar X$ and $S$.)

Then observations 3000, 3300, 3300, 3600, 3900, and 4350 (six of the nine) are within one standard deviation of the mean. (Please verify this: Find the endpoints of the interval $\bar X \pm S$. Then check which observations fall within the interval.)

So the desired proportion is $6/9 \approx 66.7\%$ (The Empirical Rule suggests this proportion is often 'about 68%`, so your percentage is pretty close. The ER is an approximate rule; don't necessarily expect exact results.)

For reference, computations from R statistical software are as follows:

x = c(2250,2250,3000,3300,3300,3600,3900,4350,5250)
a = mean(x);  s = sd(x)
a;  s
## 3466.667      # average
## 961.7692      # SD
x[abs(a-x)<=s]   # list of observations within one SD of average
## 3000 3300 3300 3600 3900 4350
pm = c(-1,1);  a + pm*s
## 2504.897 4428.436
mean(abs(x-a)<s) # proportion of obs within one SD of avg
## 0.6666667

One more thing: Now, to consolidate you understanding, please try this: The ER suggests that about 95% of the observations fall within two standard deviations of the mean. How many of your observations fall within the interval $\bar X \pm 2S ?$

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