Lecture: The Normal Distribution
Lecture script · about 15–20 minutes spoken · Week 9
Opening
Picture a bus ride downtown that usually takes about 32 minutes. Some mornings it takes 26. Some mornings it takes 38. Once in a while it drags on to 44. If you have somewhere important to be, the average alone won't save you. You need to know how often the ride runs long, and how much slack to leave.
That question shows up in a lot of jobs. A nurse asks whether one patient's reading is unusual. A transit planner decides how much slack to build into a timetable. A field biologist asks whether a bird's weight is typical for its species. All three are asking the same thing: how far is this value from the middle, measured in standard deviations, and how common is a value that far out?
Here's where we are. In Weeks 1 to 3 you learned to describe one variable: its shape, its center, its spread. Week 4 related two variables. Weeks 5 to 7 built probability, random variables, and the binomial, and Week 7 gave you a first look at the normal curve. Last week, in Week 8, you met the z-score, which says how many standard deviations a value sits above or below the mean. Today the z-score gets a curve to live on. Next week, in Week 10, Sampling Distributions & the Central Limit Theorem, you'll see why averages follow this same curve.
We'll take six ideas in order: the normal model, the empirical rule, standardizing, forward calculations, inverse calculations, and how to check whether a normal model is reasonable. Our examples come from that bus ride, a guessed quiz, and the penguins you've known since Week 1.
1. The normal model: a bell set by two numbers
Imagine you wrote down every weekday commute for a year and drew a histogram. Most bars would stand near the middle, around half an hour. The bars would shrink as you moved toward very short or very long trips. Now imagine thousands of trips and very thin bars. The jagged tops would smooth out into a curve shaped like a bell.
You met that kind of smooth curve in Week 6. It's a density curve: the total area under it is 1, and the area above a range of values is the probability of landing in that range. One family of these curves is used so often that it gets its own name. The normal distribution is the probability distribution whose density curve is the symmetric, bell-shaped normal curve. It is completely set by two numbers, its mean and its standard deviation, and a real-world variable is only ever assumed to follow it.
Those two numbers have two different jobs. The mean, which we write with the Greek letter mu, puts the peak in place. The standard deviation, the Greek letter sigma, sets how wide the bell spreads. A bigger standard deviation makes a lower, wider bell. A smaller one makes a taller, narrower bell.
Now notice the last part of that definition: "only ever assumed". No real list of commute times is exactly normal. Real trips have a fastest day and a slowest day, and sometimes a breakdown. So when we use a bell, we say "suppose". Here's the model we'll use all lecture. Suppose one-way commute times follow a normal model with a mean of 32 minutes and a standard deviation of 6 minutes. A normal model is a normal distribution with a chosen mean and standard deviation that you assume describes a real variable, always stated as an assumption.
Think of it like a recipe card. The card says "bake 32 minutes", and that's a fine plan. But you still check the oven. The model is the card, and the data are the oven.
Pause and try this. Under the commute model, is a 26-minute trip more common, less common, or equally common compared with a 38-minute trip?
Picture where each one sits on the bell.
Here's the answer. Equally common. Each trip sits 6 minutes from the mean of 32, one on each side, and the bell is symmetric, so the two are mirror images.
2. The empirical rule: 68, 95, and 99.7
Before you touch a table or a spreadsheet, a bell gives you three numbers for free. Picture the bell cut into slices, each one standard deviation wide.
The empirical rule is the rule of thumb that in a normal model about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. It fits bell-shaped data and not skewed data. Those three percents are rounded versions of exact shares, which are 0.6827, 0.9545, and 0.9973.
Let's put the rule on the bus. One standard deviation below the mean is 32 minus 6, which is 26 minutes. One above is 32 plus 6, which is 38. So about 68% of trips take between 26 and 38 minutes. Two standard deviations out, you go from 20 minutes to 44 minutes, and about 95% of trips land in there. Three out, from 14 minutes to 50 minutes, holds about 99.7%.
Here's a useful extra step. If 95% of trips land between 20 and 44 minutes, then 5% land outside. The bell is symmetric, so that 5% splits evenly: about 2.5% of trips take longer than 44 minutes. That's about 1 weekday in 40.
The rule works on counts that behave like a bell, too. In Week 7, a student guessed on 100 four-option questions. The count of right answers had a mean of 25 and a standard deviation of about 4.33. That count passed the rule of thumb, so a bell is reasonable, and about 68% of guessers score between 20.67 and 29.33 correct.
But here's the trap. The rule describes a bell, and only a bell. Think of home prices in a town with a few mansions. The long right tail stretches way out, and the left side stops short. Using 68, 95, and 99.7 there would give you the wrong picture.
Pause and try this. Under the commute model, about what percent of trips take less than 20 minutes?
Use the 95% and split what's left over.
Here's the answer. About 2.5%. A 20-minute trip sits two standard deviations below the mean. About 5% of trips fall outside 20 to 44, and half of that 5% sits on the short side.
3. Standardizing: one bell for every model
The commute model is in minutes. A model for bag weights would be in grams. A model for quiz scores would be in correct answers. It would be a pain to need a separate chart for each one. Last week's z-score fixes that.
Standardizing is turning a value into a z-score by subtracting the mean and then dividing by the standard deviation, so that values from different normal models can be read on one common scale. For a model, you take the value, subtract mu, and divide by sigma. The minutes cancel, so a z-score has no units.
Try it on a 38-minute trip. Subtract the mean: 38 minus 32 is 6. Divide by the standard deviation: 6 divided by 6 is 1. So a 38-minute trip sits 1 standard deviation above the mean. Now a 23-minute trip. Subtract: 23 minus 32 is negative 9. Divide: negative 9 over 6 is negative 1.5. That trip sits 1.5 standard deviations below the mean.
Here's the beautiful part. If you standardize every value in a normal model, the bell keeps its shape, slides its center to 0, and shrinks its spread to 1. The standard normal distribution is the normal distribution with mean 0 and standard deviation 1. Standardizing turns any normal model into this one curve, so one table of areas serves every normal model.
It's like converting prices from different countries into one currency before you compare them. Once everything is in z, every bell is the same bell.
Pause and try this. A classmate types 38 minus 32 divided by 6 into a calculator, with no parentheses, and gets a z-score far bigger than 3. What went wrong?
Think about the order of the steps.
Here's the answer. The calculator divided first: 32 over 6 is about 5.33, and then it took 5.33 away from 38. Subtract the mean first, then divide the difference by the standard deviation, and you get 1.
4. Forward: from a value to a share
Now you can answer the rider's real question. What share of trips take less than 38 minutes? That share is an area under the bell, to the left of 38. A forward normal calculation is a calculation that starts from a value and finds a proportion: standardize the value, then find the area below, above, or between under the standard normal curve.
Step one, standardize. You already did: 38 minutes is a z-score of 1. Step two, look up the area to the left of 1. You can use a table. A standard normal table is a table that lists, for each z-score, the area under the standard normal curve to the left of that z-score. That area is the proportion of a normal model below the matching value. At the row for 1.0 and the column for zero, the table gives 0.8413. So about 84.1% of trips take less than 38 minutes.
A spreadsheet can do both steps at once. Its NORM.DIST function takes the value, the mean, the standard deviation, and the word TRUE, and it returns the same 0.8413.
What about trips longer than 38 minutes? The whole area is 1, so the share above is 1 minus 0.8413, which is 0.1587, about 15.9%. That's about 16 weekdays in 100. And by symmetry, the share of trips shorter than 26 minutes is also about 15.9%, because 26 is a z-score of negative 1.
Here's the habit that keeps you safe: sketch the bell, shade the area you want, and then decide. The table always gives the area to the left. If your shaded area sits on the right, subtract from 1.
Pause and try this. A classmate reads 0.8413 at a z-score of 1 and says 84% of trips take longer than 38 minutes. What's the fix?
Ask which side of 38 the table's area sits on.
Here's the answer. The table's area sits to the left, so 84% of trips take less than 38 minutes. The share longer than 38 is 1 minus 0.8413, about 15.9%.
5. Inverse: from a share to a value
Sometimes you know the share and want the value. A transit planner doesn't ask what share of trips beat 38 minutes. The planner asks how long the timetable should allow so that 95% of trips arrive on time.
That target value is a percentile. A percentile is the value with a given percent of a distribution below it; the 95th percentile of a model is the value that 95% of the model falls below. Finding it is an inverse normal calculation: a calculation that starts from a proportion and finds a value. You find the z-score with that area to its left, then turn it back into minutes. In words: multiply the z-score by the standard deviation, and add that to the mean.
Three z-scores come up so often that they're worth knowing on sight. The z-score with 95% of the area to its left is 1.645. With 97.5% to its left, it's 1.960. With 99.5% to its left, it's 2.576.
So, the 95th percentile commute. Take 1.645 times 6, which is 9.87 minutes. Add it to 32, and you get 41.87 minutes. About 95% of trips take less than that. A spreadsheet's NORM.INV function, given 0.95, the mean, and the standard deviation, returns the same 41.87.
Now the middle 95% of trips. The middle 95% leaves 5% outside, split as 2.5% in each tail, so you need 1.960, not 1.645. Take 1.960 times 6, which is 11.76. Step down and up from 32, and the middle 95% of trips run from 20.24 to 43.76 minutes. That matches the empirical rule's 20 to 44, which used a round 2 in place of 1.960.
Keep those three z-scores, 1.645, 1.960, and 2.576, in your pocket. They hold the middle 90%, 95%, and 99% of a bell, and in Weeks 11 and 12 they return to build confidence intervals.
Pause and try this. To find the middle 95% of trips, should you look up 1.645 or 1.960?
Remember that a middle range has two tails.
Here's the answer. Use 1.960. A middle 95% leaves 2.5% in each tail, so the upper cutoff has 97.5% to its left. Using 1.645 would put all 5% in one tail.
6. Is a normal model reasonable?
Everything so far began with "suppose". Now we earn that word. Before you lean on a bell for real data, you check it, and you report a verdict of "reasonable" or "not reasonable". You never say "the data are normal", because no check can prove that.
Here are the checks. Look at the histogram's shape. Compare the mean with the median. Compare the gaps from the median to each quartile. Count the share of values within one, two, and three standard deviations. And read a normal probability plot. A normal probability plot is a graph that plots each data value against the z-score a normal model would give it at the same rank. Points that lie close to a straight line suggest a normal model is reasonable, and a steady curve suggests skew.
Let's run some checks on the 342 measured penguins. Their mean body mass is 4,201.75 grams, and their median is 4,050 grams. The mean sits above the median, pulled toward a longer right tail. Next, the quartile gaps. The median sits 500 grams above the first quartile but 700 grams below the third quartile. So the upper half of the data is more spread out than the lower half.
One check does look fine. Three standard deviations either side of the mean runs from 1,795.90 grams to 6,607.60 grams. The lightest bird, 2,700 grams, and the heaviest, 6,300 grams, both sit inside. So no bird is extreme. But passing one check isn't proof of a bell. With the mild right skew, a normal model is not reasonable for fine tail questions about all 342 penguins together. In this week's Data Lab, you'll run the same checks on one species at a time, and the verdict may change.
Pause and try this. All 342 penguin masses sit within three standard deviations of the mean. Does that make a normal model reasonable?
Think about what the other checks showed.
Here's the answer. Not on its own. The mean sits above the median and the upper quartile gap is wider, so the data lean right. One passed check isn't enough.
Closing
Let's gather it up. A normal model is a symmetric bell set by its mean and standard deviation, and you always state it as an assumption. The empirical rule gives you 68, 95, and 99.7 percent within one, two, and three standard deviations, but only on a bell. Standardizing turns any normal model into the standard normal curve. A forward calculation turns a value into an area, and the table and NORM.DIST give the area to the left. An inverse calculation turns an area into a value, with 1.645, 1.960, and 2.576 as the z-scores to know. And before you trust a bell, you check the shape, the center, the quartile gaps, and the plot, and you report "reasonable" or "not reasonable".
Next in this module come the Readings & Resources page, the lecture tutorial, the practice set, and the quiz, where you'll run these calculations yourself, and then the discussion, the assignment, and the Data Lab. Next week, in Week 10, the Central Limit Theorem shows why averages of many values follow this same bell.
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