Can you subtract variances

We always calculate variability by summing squared deviations from the mean. That gives us a variance—measured in square units square dollars, say, whatever those are. We realign the units with the variable by taking the square root of that variance. This gives us the standard deviation now in dollars again. To get the standard deviation of the sum of the variables, we need to find the square root of the sum of the squared deviations from the mean.

We add the variances, not the standard deviations. Question 2: Why add even for the difference of the variables? We buy some cereal. After all, one corn flake more or less would change the weight ever so slightly. Weights of such boxes of cereal vary somewhat, and our uncertainty about the exact weight is expressed by the variance or standard deviation of those weights. Next we get out a bowl that holds 3 ounces of cereal and pour it full.

Our pouring skill is not very precise, so the bowl now contains about 3 ounces with some variability uncertainty. How much cereal is left in the box? Well, we assume about 13 ounces. The variability of the weight in the box has increased even though we subtracted cereal. Question 2 follow-up : Okay, but is the effect exactly the same when we subtract as when we add? Suppose we have some grapefruit weighing between 16 and 24 ounces and some oranges weighing between 9 and 13 ounces.

We pick one of each at random. Question 3: Why do the variables have to be independent? Consider a survey in which we ask people two questions:. There will be some mean number of sleeping hours for the group, with some standard deviation. There will also be a mean and standard deviation of waking hours.

Clearly, variances did not add here. Why not? These data are paired, not independent, as required by the theorem. This is yet another place where students must remember to check a condition before proceeding.

Many teachers wonder if teaching this theorem is worth the struggle. I say getting students to understand this key concept 1 is not that difficult and 2 pays off throughout the course, on the AP Exam, and in future work our students do in statistics. Then they were asked:. Because rolls of the dice are independent, we can apply the Pythagorean theorem to find the variance of the total, and that gives us the standard deviation.

The CLT tells us that sums essentially the same thing as means of independent random variables approach a normal model as n increases. On the AP Exam, the investigative task asked students to consider heights of men and women. They were given that the heights of each sex are described by a normal model. Means were given as 70 inches for men and 65 inches for women, with standard deviations of 3 inches and 2. Among the questions asked was:.

Suppose a married man and a married woman are each selected at random. What is the probability the woman will be taller than the man? Because the people were selected at random, the heights are independent, so we can find the standard deviation of the difference using the Pythagorean theorem. Macro Andrew Andrew 1 1 gold badge 1 1 silver badge 6 6 bronze badges. This is, the values M1,M2, Mn are not necessarily positive. I hope this helps. Add a comment. Active Oldest Votes. Improve this answer.

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The other way around, variance is the square of SD. So: - You square the individual SD's to get the variances - Then you add these together to get the total variance - Then you take the square root to get the total SD. This works for any number of independent variables mark the bold type for independent! Key Questions If you add two independent random variables, what is the standard deviation of the combined distribution, if the standard deviations of the two original distributions were, for example, 7 and 5?

What is the procedure for calculating the new standard deviation for two combined random variables, if the random variables X and Y are not independent? Unless you know the "rules" of their dependency, you can't. If you add two independent random variables, what is the standard deviation of the combined distribution, if the standard deviations of the two original distributions were, for example, 7 and 5?

If you multiply each entry of a set by 3 and added 1, how would the mean and standard deviation change? If I know the mean, standard deviation, and size of sample A and sample B, how do I compute the standard deviation of the union of samples A and B? What if samples A and B are of different sizes?

Can you find the standard deviation of negative numbers? What is the difference between the standard deviation and margin of error?