Standard Deviation: How do I use it?

Factual Questions
1

Okay, I’m taking an online course in statistics and I understand the basics of standard deviation (SD): It’s the square root sum of the variance. I understand that 68% of the bell curve’s area will be within one SD from the mean (center), and that 95% of the area will be within two.

Question One: Can I mentally picture SD as slicing the “base” of the bell curve up into equal parts? Granted for most applications, you may or may not go past 3 SD on either side of the mean, but am I mentally picturing it correctly?

Question Two: Do you use SD, and if so, what does it tell you? Take the instance of say 100 test scores. I’ve heard that you can use SD as a way to determine ranges of scores to curve letter grades. What about stock prices, rates of return on investments, or even hockey or baseball statistics? I figure the numerically smaller your SD is, the “higher” or “more narrower/squished” your bell curve will be.

This book I’m going through is pretty dry. I can handle the “rack-and-stack” number crunching, but I think I’m missing a bigger picture.

Can you help a Doper out?

Tripler
Much obliged. I may ask more questions later. . .

2

I must admit I don’t think I’ve ever used SD outside the context of teaching it in a chemistry lab class (or probably once or twice in lab research, although I’m a biologist, and the things I was doing when I was in the lab were mostly qualitative)…

However, if you’re given a number as “mean +/- SD”, you can think of that as “the actual value is probably somewhere between mean - SD and mean + SD”. Chances are that the mean isn’t exactly on target, but the SD gives you the range of where the actual value probably lies.

Let’s go to hockey statistics to put this in context. Let’s say a player has, on average, 2 points/game, with a SD of something like 1.8. That means he will PROBABLY have between 0.2 (0) and 3.8 (4) points in a given game. If the SD were 0.1, chances would be pretty good that he’d have exactly 2 points in any given game (depending how big your sample was to computer the SD, of course).

Of course, I am NOT a statistician, so I might be full of it :slight_smile:

3

Well, let me try…

Answer One: Yes, you can think of the SD as dividing your bell curve into equal parts. Imagine your bell curve as sitting on a number line so that the mean is at zero. If your standard deviation is (say) 5, then draw lines at + and - 5, + and - 10, + and - 15 and so on. Everything between -5 and +5 is within one standard deviation and will contain about 68% of the curve. The area between -10 and +10 will contain about 95% and so on.

Answer Two: I use standard deviations quite a bit where I work. Yes, the smaller the SD the narrower (or squished) the curve is. This helps with your confidence when you are forecasting something.

Suppose you are forecasting sales (which I do, actually). Let’s suppose I’m selling fish. I get fresh fish in every day but any I don’t sell will go bad by the next day. If I buy too many I wind up throwing fish away and I lose money. But, if I don’t buy enough then customers may show up and I have no fish to sell them. I also lose money. So, how many fish should I buy?

I look through my records for the last few months and discover that I average 90 customers a day and the standard deviation is 5. This means I can be fairly certain that 68% of the time I will have between 85 and 95 sales. 95% of the time I can expect between 80 and 100. And, 99% of the time I will have between 75 and 105.

Now, what can I do with this? Well, it means that I am pretty much guarenteed to have at least 75 sales so I better have at least that much inventory on hand. Now, I am very likely to have more. The question becomes, “how much more?”

Using the mean and standard deviation, I can work out the odds for any number of expected customers. In the example here, I will get 75 or more customers 99% of the time, 76 or more 98.5% of the time and so on. 50% of the time I will have 90 or more and 1% of the time I will have 105 or more.

Now, suppose I buy fish for $10 apiece and sell them for $25. (Hey, you think fish are cheap?) I can now work out how much I can really expect to get from each fish by calculating its Expected Value.

I pretty much get my 75th customer all the time, so my 75th fish is worth $25. But, I only get the 80th customer 98% of the time so fish number 80 is only worth $24.50 to me. Customer 85 shows up around 85% of the time so I can only expect a $21.25 return on his fish. And, of course, customer 90 shows up half the time so his fish is only worth $12.50.

The 95th customer is only there 35% of the time so I can’t expect more than $8.75 (on average) for his fish. Basically I only want to buy enough fish that my expected value on the last fish is greater than $10. This means that I must have a 40% chance of selling that fish. Therefore, I shouldn’t buy more than 93 fish. Any more and I will be losing more money by throwing them away than I will by turning away customers.

So, in my case, I use Standard Deviations and they tell me how many fish to buy. Now, lets look at another of your questions, the one about grades. (Keep in mind, by the way, that what most people refer to as grading on a “curve” is really more of grading with a handicap. Most people really wouldn’t like to be graded on a curve.)

Let’s use my same numbers. You analyze the grades and find the average is 90 and the standard deviation is 5.

Since you are grading on a curve and an average is “C”, then anyone between 85 and 95 is a C. 95 to 100 is a B and anything over 100 is an A. Along the same lines, 80 to 85 is a D and anything below 80 is an F.

Using different numbers, assume a mean of 60 and a standard deviation of 15. Now, 45 to 75 is a C, 75 to 90 is a B, 90 or better is an A, 30 to 45 is a D and 30 or below is an F.

There are lots of uses for standard deviation. This is just an example. Hope it helps.

4

One way to look at the SD is that it defines how narrow the “bell” of the bell curve is. Only using the average will not tell us this. If the SD is small the bell is narrrow (most values are near the average) and if the SD is large the bell is wide (most values are some distance from the average).

If you were very good at darts and almost always could throw the dart within an inch of the bullseye, the SD willl be small. If you were bad at darts and often missed the bullseye by several inches the SD would be large. But in both cases, if you were to throw the darts many, many times, the “average” would still likely be the bullseye.

5

Oops. Just realized that I cut my fish example off too early. In actual business practice you wouldn’t want to cut off cleanly at customer 93. Since you make a certain amount of (almost) guarenteed profit on customers 1 - 75, you can use that profit to offset your expected losses on customers 94 and higher; you would probably buy enough to handle up through customer 105 and accept the loss on fishes 94 - 105 in order to keep the customers coming in. (After all, customer 98 one day may be customer 4 the next so you want them to come back that next day.) In this case the statistics let you calculate the amount of profit you can expect to make on average and can also be used to adjust your pricing (to undercut your competitor down the street and still make a profit). My example above is just the base calcuation that starts the whole process.

6

We use SD to give a shorthand presentation of how “tight” our data is. (Actually, we use Standard Error of the Mean, which is based on SD.) The greater the SD in relation to the mean, the more broadly spread out our data is. There are times when what we look for is specifically a change in SD. In that case, we can test the hypothesis by an F test of the ratios of SD.

7

My God, tanstaafl, your fish example makes it rediculously easy. The Champion Gods of Fighting Ignorance smile on you this day.

I had one or two more questions written down, I’ll post them later. I want to go through the book again and re-read a few pages to see a few things first.

Tripler
Man, am I glad I asked you guys! :smiley:

8

A simple way to look at it is your SD is an average of how far your data points will deviate from the average.

0, 150; 50, 100; and 70, 80 will all give you an average of 75, but 70, 80 will give you the smallest SD.

9

I hope it’s not a hijack to ask: what is the reasoning behind the formula picked for standard deviation? Why was one SD picked to represent (approx) 33% off mean, etc. To my amateur eyes, the definition of sd seems needlessly complex. For example, why couldn’t the average of the deviations been used instead (i.e. take each number in a sample, subtract it from the average, take the absolute value of each, and average them all)

Also, is there some physics way of visualizing what a sd is? I investigated this once and discovered that the formula for standard deviation was the same (or similar, I can’t recall) to the measurement of the momentum of a body about it’s center. (Or something like that; as I say it’s been a while and I’m far from expert). Is there some corallary between sd and a measurement in the physical world?

10

The standard deviation is the square root of the variance, which is the difference between the square of the mean and the mean of the squares. Although the standard deviation is more widely used in practice, the variance is used in theory because it has nice properties (e.g., its additive under certain conditions).

11

One thing that must be noted before we go any further on this. It only corresponds to the “classic” 2/3, 19/20 ratios (roughly) if the population really is distributed along the “normal”/“bell”/Gaussian curve. If the population distribution does not follow this model, SD becomes less immediately representative.

The method of calculating SD, if I remember my statistics history correctly, was not “chosen” to give these proportions. It just so happened that, given the density function of the Gaussian distribution, this is how it worked out.

12

Greetings fellow materials management person :smiley:
Beautiful explanation ((applause)) even touched on level of service issues.

As the size of your sample grows your numbers will self adjust. For something as perishable as the one day fish example, you would eventually build year round figures to help compensate for seasonal variations and even common daily variations like always having clams on hand on thur/fri morning for the restaraunts to make clam chowder with on friday. In the razor thin margin world of grocery stores this data can be life and death.