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Sep 26, 2018

#1
+96244
+1

Well....if I keyed everything in correctly....

Variance  = 2056.52

Std Dev   = 45.35

EDIT to correct an earlier error....

Sep 26, 2018
edited by CPhill  Sep 27, 2018
#2
+4690
+1

can you show how you got them both?

NotSoSmart  Sep 26, 2018
#3
+96244
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I've got to go now,NSS....but...I'll come back later and show you the general proceedure.....!!!

CPhill  Sep 26, 2018
#4
+4690
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Oh alright sure ok have a good day!

NotSoSmart  Sep 26, 2018
#5
+2339
+2

1) Variance measures how spread out a particular data set is. It does this by measuring the average distance each data point is from the true mean. Variance has a formula. It is the following:

$$\sigma^2=\frac{\sum_{i=1}^{N}(X_{i}-\mu)^2}{N}$$

Yes, this formula looks very complicated, but it is first important to define a few variables.

$$N = \text{sample size}\\ \mu = \text{mean of population}\\ X_i=\text{term of data}\\ \sigma^2=\text{variance}$$

What is this formula really trying to tell you to do? Well, here is an elementary breakdown:

1. Find the mean of the data.

2. Subtract each individual element from the mean

3. Square all results from the previous step and find the sum of the squared difference.

4. Divide by the sample size.

Let's get to it!

1) Find the mean of the data.

In order to find the mean, we must first find the sum of all the data points.

 $$x_i$$ $$x_1$$ 182 $$x_2$$ 274 $$x_3$$ 207 $$x_4$$ 216 $$x_5$$ 285 $$x_6$$ 190 $$x_7$$ 138 $$x_8$$ 240 $$x_9$$ 291 $$x_{10}$$ 288 $$x_{11}$$ 176 $$x_{12}$$ 186 $$x_{13}$$ 190 $$x_{14}$$ 213 $$x_{15}$$ 236 $$x_{16}$$ 242 $$x_{17}$$ 186 $$x_{18}$$ 259 $$x_{19}$$ 164 $$x_{20}$$ 172 $$x_{21}$$ 233 $$x_{22}$$ 268 $$x_{23}$$ 152 $$x_{24}$$ 274 $$\Sigma$$ 5262

If the sum of all the data points is 5262, then the mean is this number divided by the number of data points, 24.

$$\mu=\frac{5262}{24}=219.25$$

2) Subtract each individual element from the mean and square that result. I will replicate the table above.

 $$x_i$$ $$\mu$$ $$x_i-\mu$$ $$x_1$$ 182 -37.25 $$x_2$$ 274 54.75 $$x_3$$ 207 -12.25 $$x_4$$ 216 -3.25 $$x_5$$ 285 65.75 $$x_6$$ 190 -29.25 $$x_7$$ 138 -81.25 $$x_8$$ 240 20.75 $$x_9$$ 291 71.75 $$x_{10}$$ 288 68.75 $$x_{11}$$ 176 -43.25 $$x_{12}$$ 186 -33.25 $$x_{13}$$ 190 -29.25 $$x_{14}$$ 213 -6.25 $$x_{15}$$ 236 16.75 $$x_{16}$$ 242 22.75 $$x_{17}$$ 186 -33.25 $$x_{18}$$ 259 39.75 $$x_{19}$$ 164 -55.25 $$x_{20}$$ 172 -47.25 $$x_{21}$$ 233 13.75 $$x_{22}$$ 268 48.75 $$x_{23}$$ 152 -67.25 $$x_{24}$$ 274 54.75 $$\Sigma$$ 5262 219.25

3) Square all results from the previous step and find the sum of the squared difference.

 $$x_i$$ $$\mu$$ $$x_i-\mu$$ $$(x_i-\mu)^2$$ $$x_1$$ 182 -37.25 1387.5625 $$x_2$$ 274 54.75 2997.5625 $$x_3$$ 207 -12.25 150.0625 $$x_4$$ 216 -3.25 10.5625 $$x_5$$ 285 65.75 4323.0625 $$x_6$$ 190 -29.25 855.5625 $$x_7$$ 138 -81.25 6601.5625 $$x_8$$ 240 20.75 430.5625 $$x_9$$ 291 71.75 5148.0625 $$x_{10}$$ 288 68.75 4726.5625 $$x_{11}$$ 176 -43.25 1870.5625 $$x_{12}$$ 186 -33.25 1105.5625 $$x_{13}$$ 190 -29.25 855.5625 $$x_{14}$$ 213 -6.25 39.0625 $$x_{15}$$ 236 16.75 280.5625 $$x_{16}$$ 242 22.75 517.5625 $$x_{17}$$ 186 -33.25 1105.5625 $$x_{18}$$ 259 39.75 1580.0625 $$x_{19}$$ 164 -55.25 3052.5625 $$x_{20}$$ 172 -47.25 2232.5625 $$x_{21}$$ 233 13.75 189.0625 $$x_{22}$$ 268 48.75 2376.5625 $$x_{23}$$ 152 -67.25 4522.5625 $$x_{24}$$ 274 54.75 2997.5625 $$\Sigma$$ 5262 219.25 49356.5

4) Divide by the sample size

 $$x_i$$ $$\mu$$ $$x_i-\mu$$ $$(x_i-\mu)^2$$ $$\frac{(x_i-\mu)^2}{N}$$ $$x_1$$ 182 -37.25 1387.5625 $$x_2$$ 274 54.75 2997.5625 $$x_3$$ 207 -12.25 150.0625 $$x_4$$ 216 -3.25 10.5625 $$x_5$$ 285 65.75 4323.0625 $$x_6$$ 190 -29.25 855.5625 $$x_7$$ 138 -81.25 6601.5625 $$x_8$$ 240 20.75 430.5625 $$x_9$$ 291 71.75 5148.0625 $$x_{10}$$ 288 68.75 4726.5625 $$x_{11}$$ 176 -43.25 1870.5625 $$x_{12}$$ 186 -33.25 1105.5625 $$x_{13}$$ 190 -29.25 855.5625 $$x_{14}$$ 213 -6.25 39.0625 $$x_{15}$$ 236 16.75 280.5625 $$x_{16}$$ 242 22.75 517.5625 $$x_{17}$$ 186 -33.25 1105.5625 $$x_{18}$$ 259 39.75 1580.0625 $$x_{19}$$ 164 -55.25 3052.5625 $$x_{20}$$ 172 -47.25 2232.5625 $$x_{21}$$ 233 13.75 189.0625 $$x_{22}$$ 268 48.75 2376.5625 $$x_{23}$$ 152 -67.25 4522.5625 $$x_{24}$$ 274 54.75 2997.5625 $$\Sigma$$ 5262 219.25 0 49356.5 $$2056.5208\overline{3}$$

I realize that this answer differs from Cphill's. I have reviewed my work many times, and I do not believe that I have done anything incorrect.

2) The standard deviation, thankfully, does not require this much computation. Take the square root of the previous result, $$2056.5208\overline{3}$$

$$\sigma=\sqrt{2056.5208\overline{3}}\approx45.3489$$

I realize that this is a lot of information to take in at once. If you have any questions, then do not hesitate to ask.

Sep 26, 2018
#7
+96244
0

Yeah, X^2...I did this one rather rapidly.....I could have keyed in something wrong....your proceedure is correct....

CPhill  Sep 27, 2018
#6
+96244
0

OK, NSS...here's the procedure....

1.  [ Add all the data  and divide by 24]  ...this gives us the mean

2.  Using each data point....... take this data point and subtract the mean from it....square this result

For instance.... we would calculate  (182 - mean)^2  =   something

Note...that this will be a little lengthy because we must do this 24 times...once for each data point

3. Add all 24 of the values you find and divide by the number of data points (24)...... this will be the variance

4. Take the square root of  this result....this will be the standard deviation...!!!

Sep 27, 2018