8  Averages and Standard Deviation

8.1 Averages

• One summary characteristic of a distribution is the long run average value of the random variable.
• We can approximate the long run average value by simulating many values of the random variable and computing the average (mean) in the usual way.

Example 8.1

Let $X$ be the sum of two rolls of a fair four-sided die, and let $Y$ be the larger of the two rolls (or the common value if a tie). Recall your tactile simulation from Example Example 5.5. Based only on the results of your simulation, approximate the long run average value of each of the following. (Don’t worry if the approximations are any good yet.)

1. $X$
2. $Y$
3. $X^2$
4. $XY$
   
   
   
   
   
   
   
   
   
   

Example 8.2

Donny Don’t says: “Why bother creating columns for $X^2$ and $XY$? If I want to find the average value of $X^2$ I can just square the average value of $X$. For the average value of $XY$ I can just multiply the average value of $X$ and the average value of $Y$.” Do you agree? (Check to see if this works for your simulation results.) If not, explain why not.

• In general the order of transforming and averaging is not interchangeable.
• Whether in the short run or the long run, in general $$\begin{array}{rl}\text{Average of }g(X)& \ne g(\text{Average of }X)\\ \text{Average of }g(X,Y)& \ne g(\text{Average of }X,\text{Average of }Y)\end{array}$$

Example 8.3

Recall your tactile simulation from Example Example 5.5. Let $U_1$ be the result of the first roll, and $U_2$ the result of the second, so the sum is $X=U_1+U_2$.

1. Donny Don’t says: “$X=U_1+U_2$, so I can find the average value of $X$ by finding the average value of $U_1$, the average value of $U_2$, and adding the two averages”. Do you agree? Explain.
   
   
   
   
   
2. Donny Don’t says: “$U_1$ and $U_2$ have the same distribution, so they have the same average value, so I can find the average value of $X$ by multiplying the average value of $U_1$ by 2”. Do you agree? Explain.
   
   
   
   
   
3. Donny Don’t says: “$U_1$ and $U_2$ have the same distribution, so $X=U_1+U_2$ has the same distribution as $2U_1=U_1+U_1$”. Do you agree? Explain.
   
   
   
   
   

• In general the order of transforming and averaging is not interchangeable.
• However, the order is interchangeable for linear transformations.
• Linearity of averages.: If $X$ and $Y$ are random variables and $a$ and $b$ are non-random constants, whether in the short run or the long run, $$\begin{array}{rl}\text{Average of }a+bX& =a+b(\text{Average of }X)\\ \text{Average of }X+Y& =\text{Average of }X+\text{Average of }Y\end{array}$$
• The average of the sum of $X$ and $Y$ is the sum of the average of $X$ and the average of $Y$ regardless of the relationship between $X$ and $Y$.

8.2 Standard deviation

• The long run average value is just one feature of a distribution.
• Random variables vary, and the distribution describes the entire pattern of variability.
• We can summarize this degree of variability by listing some percentiles (10th, 25th, 50th, 75th, 90th, etc); the more percentiles we provide the clearer the picture, but the less of a summary.
• Roughly, standard deviation measures overall degree of variability in ai single number, as the average distance from the mean.
• Technically, the variance is the long run average squared distance from the mean, and the standard deviation is the square root of the variance.

$$\begin{array}{rl}\text{Variance of }X& =\text{Average of }((X-\text{Average of }X{)}^2)\\ \text{Standard deviation of }X& =\sqrt{\text{Variance of }X}\end{array}$$

Example 8.4

We’ll compare long run average and standard deviation for the Uniform(0, 60) distribution and the Normal(30, 10) distribution. A Normal(30, 10) distribution mean (long run average) 30 and standard deviation 10.

1. Make an educated guess for the long run average value of a Uniform(0, 60) distribution. Explain.
   
   
   
   
   
2. Will the standard deviation for a Uniform(0, 60) distribution be greater than, less than, or equal to 10, the standard deviation for a Normal(30, 10) distribution? Explain without doing any calculations.
   
   
   
   
   
3. Make an educated guess for the standard deviation of a Uniform(0, 60) distribution.
   
   
   
   
   

• Standard deviation provides a “ruler” by which we can judge a particular realized value of a random variable relative to the distribution of values.
• Standardization measures values in terms of “standard deviations away from the mean”
• This idea is particularly useful when comparing random variables with different measurement units but whose distributions have similar shapes.

$$\text{Standardized value}=\frac{\text{Value - Mean}}{\text{Standard deviation}}$$

Example 8.5

SAT scores have, approximately, a Normal distribution with a mean of 1050 and a standard deviation of 200. ACT scores have, approximately, a Normal distribution with a mean of 21 and a standard deviation of 5.5. Darius’s score on the SAT is 1500. Alfred’s score on the ACT is 31. Who scored relatively better on their test?

1. Compute the deviation from the mean for Darius’s SAT score. How does this compare to the average deviation from the mean for SAT scores?
   
   
   
   
   
2. Compute the deviation from the mean for Alfred’s ACT score. How does this compare to the average deviation from the mean for ACT scores?
   
   
   
   
   
3. Who scored relatively better on their test?
   
   
   
   
   

• Any Normal distribution follows the “empirical rule” which determines the percentiles that give a Normal distribution its particular bell shape.
• For a Normal distribution:
  • 68% of values are within 1 standard deviation of the mean
  • 95% of values are within 2 standard deviations of the mean
  • 99.7% of values are within 3 standard deviations of the mean
• The table below lists some percentiles of a Normal distribution.
• The “standard” Normal distribution is a Normal(0, 1) distribution, with a mean 0 and a standard deviation of 1.

Empirical rule for Normal distributions

Percentile	SDs away from the mean
0.1%	3.09 SDs below the mean
0.5%	2.58 SDs below the mean
1%	2.33 SDs below the mean
2.5%	1.96 SDs below the mean
10%	1.28 SDs below the mean
15.9%	1 SDs below the mean
25%	0.67 SDs below the mean
30.9%	0.5 SDs below the mean
50%	0 SDs above the mean
69.1%	0.5 SDs above the mean
75%	0.67 SDs above the mean
84.1%	1 SDs above the mean
90%	1.28 SDs above the mean
97.5%	1.96 SDs above the mean
99%	2.33 SDs above the mean
99.5%	2.58 SDs above the mean
99.9%	3.09 SDs above the mean