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Week 9 · Chapter

Chapter 9 — The Normal Distribution

Introduction to Statistics Free sample edition

The module's primary reading · ~35–45 minutes · Objective 5 (use the normal model and sampling distributions, including the Central Limit Theorem, to reason about variability in individual values and in sample statistics) · Read it first this week; the lecture, the tutorial, the quiz, and the Data Lab all build on it.

In plain language

Many measurements pile up in the middle and thin out evenly on both sides, like a bell. This week you learn the normal distribution (the bell-shaped model set by a center and a spread) and how to use it to answer "how common is that?" You will learn a quick rule of thumb for the bell, and a way to put any value on one common scale. You will find the share of values below a cutoff, and the cutoff that matches a share. Then you will learn to check whether a bell is a fair description of real data before you lean on it.

Why you already care about this

A transit timetable says the ride downtown takes about 32 minutes. Some mornings it takes 26 minutes, some mornings 38, and once in a while 44. If you have a 9 o'clock appointment, the average alone is not enough. You want to know how often the trip runs long, and how much slack to leave so you are late only rarely.

Questions like that come up everywhere. A nurse asks whether one patient's reading is unusual. A bakery asks how many loaves come out too light. A field biologist asks whether a bird's weight is typical for its species. Last week you met the z-score, the number of standard deviations a value sits from the mean. This week the z-score gets a curve to live on, and that curve turns "how far from the middle" into "how common".

By the end of this chapter you will be able to:

  • describe a normal model by its mean and standard deviation, and state it as an assumption;
  • use the empirical rule to give the share of values within one, two, or three standard deviations;
  • standardize a value with a z-score and explain the standard normal distribution;
  • find the share of a normal model below, above, or between values, by table and by spreadsheet;
  • find the value that cuts off a given share, by table and by spreadsheet;
  • judge whether a normal model is reasonable for a set of data.

Carry this sentence with you: a normal model is an assumption you check — not a fact.

Where we are

  • Already covered: Weeks 1 to 3 described one variable, Week 4 related two, Weeks 5 to 7 built probability, random variables, and the binomial, and Week 8 added the z-score.
  • This week: the normal distribution: the normal model, the empirical rule, standardizing, forward and inverse normal calculations, and checking whether a normal model is reasonable.
  • How they connect: a z-score measures distance from the mean, and the normal curve turns that distance into a share of values.
  • Where we're going next: Week 10, Sampling Distributions & the Central Limit Theorem, shows why averages of many values follow this same bell.

Each earlier week feeds this one. Week 2 taught you to read the shape of a histogram and to name skew by its tail. Week 3 gave you the mean, the median, the standard deviation, and the quartiles, which you use again in Section 6. Week 6 introduced a density curve: a smooth curve where the total area underneath is 1 and the area above a range of values is the probability of landing there. Week 7 gave you a first look at the normal curve as a model for many measurements. Week 8 taught you the z-score, z = (x − x̄) ⁄ s, as relative standing. This week those pieces snap together.

1. The normal model: a bell set by two numbers

Picture the commute times for one month of weekday trips drawn as a histogram. Most bars stand near the middle, around half an hour. The bars shrink as you move toward very short or very long trips. Now imagine thousands of trips and very thin bars. The jagged tops smooth out into a curve.

That smooth curve is a density curve, and the area under it plays the role of the bars. The share of trips between 30 and 35 minutes is the area under the curve between 30 and 35. The whole area is 1, which stands for 100% of trips.

One family of density curves is used so often that it has 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 μ (the center) and its standard deviation σ (the spread), and a real-world variable is only ever assumed to follow it.

The two numbers do two different jobs. The mean μ (read "mu") sets where the peak sits. The standard deviation σ (read "sigma") sets how wide the bell spreads. A larger σ gives a lower, wider bell; a smaller σ gives a taller, narrower one. The total area stays 1 either way.

  • 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 (for example, suppose commute times follow a bell with mean 32 minutes and standard deviation 6 minutes).

That last phrase matters. No real list of commute times is exactly normal. Real data have gaps, ties, and a longest trip. A normal model is a useful simplification you choose, so you say "suppose" every time.

Worked example — the commute-time normal model. Here is the model you will use all week.

  1. The variable. Let X (a random variable that follows the model) be one commute time, in minutes.
  2. The assumption. Suppose one-way commute times X follow a normal model with μ = 32 minutes and σ = 6 minutes. This is an assumption you choose to make, not a claim about any real city.
  3. The picture in words. The bell peaks at 32 minutes. It is symmetric, so a trip 6 minutes shorter than average (26 minutes) is exactly as common as a trip 6 minutes longer (38 minutes).

In words: under this model, a typical trip takes about 32 minutes, give or take about 6. A rider who plans around 32 minutes alone will be late about half the time, because half the area sits above the mean.

Misconception check — calling real data "normal". Students often say "commute times are normal" as if it were a fact about the world. It is tempting, because the bell fits many measurements well. But a model is a tool, not a finding. The cure: write "suppose commute times follow a normal model with mean 32 minutes and standard deviation 6 minutes", and keep the word suppose. Section 6 shows you how to check whether the tool fits.

Memory hook: the mean places the bell; the standard deviation sets its width.

2. The empirical rule: read 68, 95, and 99.7 off the bell

Before any table or spreadsheet, a normal model gives you three numbers for free. Think of the bell as 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.

The rule is a rounded version of exact shares. The exact constants are 0.6827, 0.9545, and 0.9973, for one, two, and three steps out from the mean. The empirical rule rounds those to 68%, 95%, and 99.7%.

Worked example — the empirical rule on the commute model. Use μ = 32 minutes and σ = 6 minutes.

  1. One standard deviation. μ − σ = 32 − 6 = 26 minutes and μ + σ = 32 + 6 = 38 minutes. So about 68% of trips take between 26 and 38 minutes.
  2. Two standard deviations. 32 − 2 × 6 = 20 minutes and 32 + 2 × 6 = 44 minutes. So about 95% of trips take between 20 and 44 minutes.
  3. Three standard deviations. 32 − 3 × 6 = 14 minutes and 32 + 3 × 6 = 50 minutes. So about 99.7% of trips take between 14 and 50 minutes.
  4. A tail. About 100% − 95% = 5% of trips fall outside 20 to 44 minutes, and symmetry splits that evenly. So about 2.5% of trips take longer than 44 minutes.

In words: on about 1 weekday in 40, the ride runs past 44 minutes. A rider who leaves 44 minutes before an appointment will be late on roughly 1 trip in 40.

The same rule works on any bell, whatever its units. Suppose the fill weights of rice bags from one packing machine follow a normal model with mean 1,000 g and standard deviation 5 g. Then about 95% of bags weigh between 1,000 − 2 × 5 = 990 g and 1,000 + 2 × 5 = 1,010 g. A quality inspector who sees a 985 g bag knows it sits three standard deviations below the mean, out where only a sliver of bags should fall.

You also met a count that behaves like a bell. In Week 7, guessing on 100 four-option questions gave a count with mean 25 and standard deviation about 4.33 correct answers. The rule of thumb from that week held: np = 25 and n(1 − p) = 75 are both at least 10. So a normal model is reasonable here, and about 68% of guessers score between 25 − 4.33 = 20.67 and 25 + 4.33 = 29.33 correct.

Misconception check — the empirical rule on a skewed list. Students often apply 68, 95, and 99.7 to any data set at all. The numbers are easy, so it is tempting. But the rule describes a bell. On a right-skewed set, such as home prices with a few mansions, the long tail stretches far past two standard deviations on one side and stops short on the other. The cure: look at the shape first. Use the empirical rule only when the histogram is roughly symmetric and single-peaked.

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

  • (a) Think of a trip you make often, by bus, car, bike, or on foot. Would a symmetric bell describe your trip times, or is there a long tail on one side? Say what causes the tail.
  • (b) Think of something you buy that is filled or cut by a machine, such as cereal, coffee, or bread. Why might its weights pile up in a bell around the label value?

Memory hook: 68, 95, 99.7 — and only on a bell.

3. Standardizing: turn any normal model into one curve

The commute model is measured in minutes, the rice model in grams, and the quiz model in correct answers. It would be a pain to need a different chart for every one. The fix is the z-score you learned last week.

  • 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 write z = (x − μ) ⁄ σ, where x is one particular value. The z-score says how many standard deviations x sits above (positive z) or below (negative z) the mean. It has no units, because the minutes on top cancel the minutes on the bottom.

When you standardize every value of a normal model, the bell keeps its shape. Its center slides to 0, and its spread becomes 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.

We use the capital letter Z for a random variable that follows the standard normal distribution.

Worked example — two commute times on the z scale. Use μ = 32 minutes and σ = 6 minutes.

  1. A long trip. For a 38-minute trip, z = (x − μ) ⁄ σ = (38 − 32) ⁄ 6 = 6 ⁄ 6 = 1. So a 38-minute trip sits 1 standard deviation above the mean.
  2. A short trip. For a 23-minute trip, z = (23 − 32) ⁄ 6 = −9 ⁄ 6 = −1.5. So a 23-minute trip sits 1.5 standard deviations below the mean.

In words: the 38-minute trip is one step of 6 minutes above the middle, and the 23-minute trip is one and a half steps below it. A rider comparing a slow bus day with a slow day on the bike can now compare z-scores instead of minutes.

Misconception check — dividing first. Students often compute 38 − 32 ⁄ 6 and get 38 − 5.33, which is far too big to be a z-score. Order of operations is the trap: division happens before subtraction unless parentheses say otherwise. The cure: subtract the mean first, inside parentheses, then divide the difference by the standard deviation. A z-score for a commute time should almost always land between −3 and 3.

Memory hook: subtract the center, divide by the spread — one bell for all.

4. Forward calculations: from a value to a proportion

Now you can settle the question a rider cares about. 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.

You write P(X < x) for the proportion of the model below the value x, and P(Z < z) for the area under the standard normal curve to the left of z. Standardizing turns the first into the second: P(X < 38) is the same area as P(Z < 1).

To read that area, 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, which is the proportion of a normal model below the matching value.

Most tables list z to two decimals. The row gives the first decimal, such as 1.0, and the column gives the second, such as 0.00. Where the row and the column meet, you read the area.

Worked example — trips shorter and longer than 38 minutes. Use μ = 32 minutes and σ = 6 minutes.

  1. Standardize. For x = 38, z = (38 − 32) ⁄ 6 = 1.
  2. Read the table. At row 1.0 and column 0.00, the table gives 0.8413. So P(X < 38) = P(Z < 1), which the standard normal table pins at 0.8413.
  3. Say it in words. About 84.1% of trips take less than 38 minutes.
  4. Flip to "above" with the complement rule. The whole area is 1, so P(X > 38) = 1 − 0.8413 = 0.1587. About 15.9% of trips take longer than 38 minutes.
  5. Use symmetry for the low side. A 26-minute trip has z = (26 − 32) ⁄ 6 = −1. The bell is symmetric about its mean, so the area below −1 matches the area above 1. So P(X < 26) is also about 15.9%.

In words: on about 16 weekdays in 100, the ride runs past 38 minutes. If being 6 minutes late costs you a missed connection, you now know how often that will happen.

You might notice that the area between 26 and 38 minutes is the 68% from the empirical rule. Subtracting the two rounded table entries gives a number that is off in the last decimal place. That is only rounding: the exact share is 0.6827.

In practice (your tools). A spreadsheet does the standardizing and the table lookup in one step. In Google Sheets or Excel, type =NORM.DIST(38, 32, 6, TRUE) in any cell. The four inputs are the value, the mean, the standard deviation, and TRUE for "area to the left". It returns 0.8413 to four decimals; use Format, then Number, to show four decimals in Sheets, or the Decrease Decimal button in Excel. For the area above 38, type =1 - NORM.DIST(38, 32, 6, TRUE). If you already have a z-score, =NORM.S.DIST(1, TRUE) reads the standard normal curve directly and gives the same 0.8413.

Honesty note: the spreadsheet answer is only as good as the model you typed in. If commute times are really skewed, a precise-looking 0.8413 can still be wrong.

Misconception check — reading the table as "area above". Students often read 0.8413 at z = 1 and say "84% of trips take longer than 38 minutes". It is tempting because 38 is the bigger number. But the table always gives the area to the left, below the value. The cure: sketch the bell, shade the area you want, and ask whether it sits left or right of the value. If it sits right, subtract the table area from 1.

Memory hook: the table reads left — shade first, then subtract for the right.

5. Inverse calculations: from a proportion to a value

Sometimes the question runs the other way. A transit planner does not ask "what share of trips take less than 38 minutes?" The planner asks "how long should the timetable allow so that 95% of trips arrive on time?" Now you know the share and want the value.

  • The 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.
  • An inverse normal calculation is a calculation that starts from a proportion and finds a value: find the z-score with that area to its left, then turn it back into the variable's units with x = μ + z · σ.

The second formula is the z-score formula solved for x. Multiply the z-score by the standard deviation to get a distance in minutes, then add that distance to the mean.

Three z-scores come up so often that you should know them on sight. The z-score with 95% of the area to its left is 1.645. The z-score with 97.5% to its left is 1.960. The z-score with 99.5% to its left is 2.576. In the table, you search the body for the area and read the z from the row and the column. For 0.95, the closest entries sit halfway between z = 1.64 and z = 1.65, which is why the percentile is written 1.645.

Worked example — the 95th percentile commute. Use μ = 32 minutes and σ = 6 minutes.

  1. Find z. The z-score with 95% of the area to its left is 1.645.
  2. Turn z into minutes. x = μ + z · σ = 32 + 1.645 × 6 = 32 + 9.87 = 41.87 minutes, rounded to two decimals.
  3. Say it in words. About 95% of trips take less than 41.87 minutes.

In words: a planner who allows about 42 minutes will see roughly 95 trips in 100 arrive on time, and about 5 in 100 run late. A rider with an appointment can use the same number to decide when to leave.

Worked example — the middle 95% of trips. Now you want a range in the middle, not a cutoff on one side.

  1. Split the tails. The middle 95% leaves 5% outside, so 2.5% sits in each tail. The z-scores ±1.960 cut off 2.5% in each tail.
  2. Find the distance. 1.960 × 6 = 11.76 minutes.
  3. Step down and up from the mean. The middle 95% of trips run from 32 − 11.76 = 20.24 to 32 + 11.76 = 43.76 minutes.

In words: about 95 trips in 100 take between about 20 and 44 minutes. That matches the empirical rule's 20 to 44 minutes, which used 2 in place of 1.960.

The same idea gives other middle shares. The z-scores ±1.645 hold the middle 90% of a normal model, and ±2.576 hold the middle 99%. Keep these three in mind. In Weeks 11 and 12 they come back as the critical values for 90%, 95%, and 99% confidence intervals.

In practice (your tools). In Google Sheets or Excel, =NORM.INV(0.95, 32, 6) returns the 95th percentile directly: 41.87 minutes, rounded to two decimals. The three inputs are the area to the left, the mean, and the standard deviation. For the bottom of the middle 95%, type =NORM.INV(0.025, 32, 6); for the top, =NORM.INV(0.975, 32, 6). To get a z-score instead of minutes, =NORM.S.INV(0.95) reads the standard normal curve and returns about 1.645.

Misconception check — using the wrong area for a middle range. Students often look up 0.95 to find the middle 95% and use z = 1.645. That puts 5% in one tail and none in the other, so the range is lopsided. The cure: a middle share leaves its leftover split between two tails. For the middle 95%, look up 0.975 (that is, 95% plus 2.5%) and use z = 1.960 on both sides.

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

  • (a) A timetable could be built around the 90th, 95th, or 99th percentile of trip times. Which would you choose for a bus that riders use to get to work, and what does each choice cost the riders or the transit agency?
  • (b) A clinic wants to flag readings that are unusually high. Should it flag the top 5% or the top 1% of a normal model? Explain what goes wrong with each choice.

Memory hook: forward turns a value into an area; inverse turns an area into a value.

6. Checking the model: decide whether a normal model is reasonable

Every calculation so far started with "suppose". This section is about earning that word. Before you use a normal model on real data, you check it, and you report a verdict of "reasonable" or "not reasonable". You never report "the data are normal", because no check can prove that.

Here are the checks, from quickest to most careful.

  1. Look at the histogram. Is it single-peaked and roughly symmetric, or does one tail stretch out?
  2. Compare the mean with the median. In a bell they sit close together. Skew pulls the mean toward the long tail.
  3. Compare the quartile gaps. In a bell, the median sits about halfway between Q1 and Q3. If one gap is much wider, the data lean that way.
  4. Run the empirical-rule check. Count the share of values within one, two, and three standard deviations of the mean, and compare with 68%, 95%, and 99.7%.
  5. 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.

You do not need to draw a perfect line. Ask whether the points follow a line with small wiggles, or bend steadily away from it at one end. With the data values on the vertical axis, a right-skewed set bends upward at the right end, because its largest values sit farther above the middle than a bell expects.

Worked example — the penguin body masses. You know these 342 measured birds from earlier weeks. Run the checks with the pinned summaries.

  1. Mean versus median. The 342 measured body masses have x̄ = 4,201.75 g above the median of 4,050 g, so the mean is pulled toward a longer right tail. In Week 2 you saw the same thing in the histogram: a mild right skew.
  2. Quartile gaps. The median sits 4,050 − 3,550 = 500 g above Q1 but 4,750 − 4,050 = 700 g below Q3. So the upper half of the data is more spread out than the lower half: 700 g versus 500 g, a ratio of 1.4.
  3. Three standard deviations. With s = 801.95 g, x̄ − 3s = 4,201.75 − 3 × 801.95 = 1,795.90 g and x̄ + 3s = 4,201.75 + 3 × 801.95 = 6,607.60 g. The lightest bird (2,700 g) and the heaviest (6,300 g) both sit inside that range.
  4. Verdict. The mild right skew means a normal model is not reasonable for fine tail questions about all 342 penguins together, even though no value sits beyond three standard deviations.

In words: if a researcher used a normal model to say what share of these penguins weigh over 5,500 g, the answer would rest on a shape the data do not quite have. Passing one check is not proof of a bell. In this week's Data Lab, you run the same checks on one species at a time, where the story can change.

Honesty note: a check can only say a model is reasonable or not; that is its limitation. Two analysts can look at the same mild skew and disagree about whether it matters, so say which checks you ran and what you saw.

In practice (your tools). To run the empirical-rule check in Google Sheets or Excel, put one group's values in column A; the formulas below assume the 151 Adelie body masses sit in A2:A152, as they will in the Data Lab. Find the mean with =AVERAGE(A2:A152) in cell D1 and the standard deviation with =STDEV(A2:A152) in cell D2. Then =COUNTIFS(A2:A152, ">="&(D1-D2), A2:A152, "<="&(D1+D2)) counts the values within one standard deviation; divide by the number of values to get the share. For a normal probability plot, sort column A from smallest to largest, then in B2 type =NORM.S.INV((ROW(A2)-1.5)/151) and fill it down beside all 151 values. Choose Insert, then Chart, then Scatter chart, with column B across and column A up.

Misconception check — "it passed one check, so it is normal". Students often run the empirical-rule check, see shares near 68% and 95%, and declare the data normal. It is tempting because the numbers line up. But the 342 penguin masses keep every value within three standard deviations and are still skewed. The cure: run more than one check, and report a verdict of "reasonable" or "not reasonable" for the question at hand, never "the data are normal".

Memory hook: the verdict is "reasonable" — not "normal".

Key terms

  • normal distribution — The probability distribution whose density curve is the symmetric, bell-shaped normal curve; it is completely set by two numbers, its mean μ (the center) and its standard deviation σ (the spread), and a real-world variable is only ever assumed to follow it. (first used: 1. The normal model: a bell set by two numbers)
  • normal model — A normal distribution with a chosen mean and standard deviation that you assume describes a real variable, always stated as an assumption (for example, suppose commute times follow a bell with mean 32 minutes and standard deviation 6 minutes). (first used: 1. The normal model: a bell set by two numbers)
  • empirical rule — 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. (first used: 2. The empirical rule: read 68, 95, and 99.7 off the bell)
  • standardizing — 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. (first used: 3. Standardizing: turn any normal model into one curve)
  • standard normal distribution — 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. (first used: 3. Standardizing: turn any normal model into one curve)
  • forward normal calculation — 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. (first used: 4. Forward calculations: from a value to a proportion)
  • standard normal table — A table that lists, for each z-score, the area under the standard normal curve to the left of that z-score, which is the proportion of a normal model below the matching value. (first used: 4. Forward calculations: from a value to a proportion)
  • percentile — 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. (first used: 5. Inverse calculations: from a proportion to a value)
  • inverse normal calculation — A calculation that starts from a proportion and finds a value: find the z-score with that area to its left, then turn it back into the variable's units with x = μ + z · σ. (first used: 5. Inverse calculations: from a proportion to a value)
  • normal probability plot — 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. (first used: 6. Checking the model: decide whether a normal model is reasonable)

Summary

  • A normal model is a symmetric bell set by its mean and its standard deviation, and you always state it as an assumption.
  • In a bell, about 68% of values sit within one standard deviation of the mean, about 95% within two, and about 99.7% within three.
  • The empirical rule describes bells only; on skewed data it misleads.
  • Standardizing subtracts the mean and divides by the standard deviation, which moves every normal model onto the standard normal curve.
  • A forward calculation turns a value into an area; the table and NORM.DIST give the area to the left.
  • An inverse calculation turns an area into a value: find z, then add z standard deviations to the mean.
  • The z-scores 1.645, 1.960, and 2.576 hold the middle 90%, 95%, and 99% of a bell.
  • Before you lean on a normal model, check the histogram, the mean and median, the quartile gaps, the empirical rule, and a normal probability plot, then report "reasonable" or "not reasonable".

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

  1. Suppose the weights of loaves from one bakery oven follow a normal model with mean 500 g and standard deviation 10 g. Between what two weights do about 95% of loaves fall? About what share of loaves weigh more than 510 g?
  2. Suppose reaction times on a sports-science tapping test follow a normal model with mean 250 milliseconds and standard deviation 20 milliseconds. Find the z-score for a reaction time of 210 milliseconds and say in words what it means.
  3. Suppose the lengths of rides on a city bike-share follow a normal model with mean 18 minutes and standard deviation 4 minutes. How long is the 95th percentile ride? Give the spreadsheet formula that checks your answer.
  4. A town's home sale prices have a mean far above their median. A classmate uses the empirical rule to say that about 95% of homes sold within two standard deviations of the mean price, split evenly on both sides. What would you tell the classmate?

Answers (worked reasoning — read after attempting)

  1. Two standard deviations is 2 × 10 = 20 g. So about 95% of loaves fall between 500 − 20 = 480 g and 500 + 20 = 520 g. For "more than 510 g", standardize: z = (510 − 500) ⁄ 10 = 1. The table's area to the left of z = 1 is 0.8413, so the share above is 1 − 0.8413 = 0.1587, about 15.9% of loaves. A baker who sells loaves over 510 g as "large" would get about 16 of every 100.
  2. Subtract the mean first, then divide: z = (210 − 250) ⁄ 20 = −40 ⁄ 20 = −2. The reaction sits 2 standard deviations below the mean, so it is unusually fast. By the empirical rule, only about 2.5% of reactions are faster than that.
  3. The 95th percentile has z = 1.645. So x = 18 + 1.645 × 4 = 18 + 6.58 = 24.58 minutes, rounded to two decimals. About 95% of rides are shorter than about 24.6 minutes. The formula =NORM.INV(0.95, 18, 4) checks it.
  4. A mean far above the median signals a right skew, probably from a few very expensive homes. The empirical rule describes a bell, so its 95% and its even split do not apply here. The long right tail can stretch past two standard deviations above the mean, while the left side stops much closer to it. Look at the histogram and run the checks first; the verdict here is likely "a normal model is not reasonable".

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, and the quiz — and next week the Central Limit Theorem explains why averages follow this normal model.

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