Distributions & the Normal Curve
See the shape of your data: the symmetric bell curve, skewed shapes, the 68-95-99.7 rule, and how to read where one value falls.
A distribution is the shape of a whole data set: it tells you where most values pile up, how spread out they are, and whether they lean to one side. Many natural measurements (heights, blood pressures, reaction times, test scores) stack up into a symmetric bell shape called the normal curve, and once you know a data set is roughly normal you can predict a lot from just two numbers, the mean and the standard deviation. The 68-95-99.7 rule says about 68% of values sit within one standard deviation of the mean, about 95% within two, and about 99.7% within three. Epidemiologists use distributions to see whether a symptom count is unusually high, biostatisticians use the normal curve to decide which lab results fall outside the healthy range, lab scientists use it to flag an outlier measurement, and doctors reading a study use percentiles (a baby at the 25th percentile for weight) to place one patient against a whole population. Learn to read the shape and you can tell an ordinary value from a surprising one at a glance.
- Common Core · HSS-ID.A.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages using the empirical (68-95-99.7) rule.
- Common Core · HSS-ID.A.1Represent data with plots on the real number line and describe the shape of a distribution, including symmetry and skew.
- Ohio · Ohio HS S.ID.4Recognize that the empirical rule applies only to normal (bell-shaped) distributions and use the mean and standard deviation to estimate percentages of data within one, two, and three standard deviations.
- NGSS · SEP-4Analyzing and Interpreting Data: describe the shape, center, and spread of a data set and use it to decide whether a single value is typical or unusual.
- AP · AP Bio SP 6 (Quantitative)Analyze and interpret quantitative data, including using the normal distribution and standard deviation to describe biological variation.
- Find the mean (average) of a data set: The normal curve is centered on the mean, so students must be able to locate the average before they can read the curve.
- Understand spread and standard deviation: The empirical rule counts values by how many standard deviations they sit from the mean, so students need the idea of spread.
- Read a percent as part of a whole: 68%, 95%, and percentiles are all parts of the full data set, so students must interpret a percent of the total.
Prerequisites are inferred: pending teacher review.
Re-learn the skill with worked practice and clear examples.
For a normal curve, the 68-95-99.7 (empirical) rule turns the mean and standard deviation into predictions. About 68% of values fall within 1 SD of the mean, about 95% within 2 SD, and about 99.7% within 3 SD. Mark the mean, step out by whole standard deviations, and read off the percentages.
IQ scores are roughly normal with a mean of 100 and a standard deviation of 15. Between which two scores do about 68% of people fall?
Reviewed- A.Between 85 and 115
- B.Between 70 and 130
- C.Between 55 and 145
- D.Between 95 and 105
Show the worked solution ▾
Answer: A. Between 85 and 115
- Step 1: 68% means within 1 SD: The empirical rule puts about 68% of values within one standard deviation of the mean.
- Step 2: Step out one SD: 100 - 15 = 85 and 100 + 15 = 115, so about 68% fall between 85 and 115.
Why it's right: About 68% of a normal distribution lies within 1 SD of the mean: 100 minus 15 is 85 and 100 plus 15 is 115.
- B: 70 to 130 is within 2 SD, which holds about 95%, not 68%.
- C: 55 to 145 is within 3 SD, which holds about 99.7%.
- D: 95 to 105 is only part of a standard deviation, far less than 68%.
Aligned to Common Core HSS-ID.A.4: estimate percentages with the empirical rule · reading level ~grade 9
Using the marked normal curve, a value sits exactly at +1 SD. About what percent of the data is at or below that value?
Reviewed- A.34%
- B.50%
- C.68%
- D.84%
Show the worked solution ▾
Answer: D. 84%
- Step 1: Below the mean is half: A normal curve is symmetric, so exactly 50% of the data sits below the mean.
- Step 2: Add the slice from mean to +1 SD: The 68% between -1 SD and +1 SD splits evenly, so 34% sits between the mean and +1 SD.
- Step 3: Total at or below +1 SD: 50% (below the mean) + 34% (mean to +1 SD) = 84%.
Why it's right: 50% of a normal curve is below the mean, and half of the 68% band (34%) lies between the mean and +1 SD, so 50 + 34 = 84% is at or below +1 SD.
- A: 34% is only the slice between the mean and +1 SD, not the whole area below it.
- B: 50% is just the part below the mean; it leaves out the mean-to-+1-SD slice.
- C: 68% is the band between -1 SD and +1 SD, not the area below +1 SD.
Aligned to Common Core HSS-ID.A.4: use the empirical rule to find a percentile · reading level ~grade 9
SAT section scores are roughly normal with a mean of 500 and a standard deviation of 100. Between which two scores do about 95% of test takers fall?
Reviewed- A.Between 400 and 600
- B.Between 300 and 700
- C.Between 200 and 800
- D.Between 450 and 550
Show the worked solution ▾
Answer: B. Between 300 and 700
- Step 1: 95% means within 2 SD: The empirical rule puts about 95% of values within two standard deviations of the mean.
- Step 2: Step out two SD: Two SD is 2 x 100 = 200. So 500 - 200 = 300 and 500 + 200 = 700.
Why it's right: About 95% of a normal distribution lies within 2 SD of the mean: 500 minus 200 is 300 and 500 plus 200 is 700.
- A: 400 to 600 is within 1 SD, which holds about 68%, not 95%.
- C: 200 to 800 is within 3 SD, which holds about 99.7%.
- D: 450 to 550 is only within half a standard deviation, far less than 95%.
Aligned to Ohio HS S.ID.4: estimate the 95% (2 SD) interval · reading level ~grade 9
- A nurse sees a patient's resting heart rate is more than 2 SD above the mean and flags it as unusually high.
- A teacher tells a class that scores within 1 SD of the average (about 68% of them) are the typical range for the test.
- A quality-control scientist rejects a batch when a measurement lands beyond 3 SD, since that is almost never normal variation.
Fill these in as you work through the lesson.
- Distribution (the shape of the whole data set):
- Normal (bell) curve (symmetric, peak at the mean):
- Standard deviation (how spread out the data is):
- Percentile (percent of data at or below a value):
For a normal curve, about % of values fall within 1 standard deviation of the mean, about % within 2 standard deviations, and about % within 3 standard deviations.
- How is a right-skewed shape different from a normal curve, and why does the empirical rule not apply to it?
- If the mean is 100 and the SD is 15, what two values mark 1 SD on each side of the mean?
- What does it mean for a value to be at the 84th percentile of a normal curve?
A test is normal with mean 70 and SD 8. Within 2 SD of the mean sits about 95% of scores. Compute the two boundaries: 70 - 2 x 8 = ____ and 70 + 2 x 8 = ____ , so about 95% of scores fall between those two numbers.
The vocabulary of this topic, shown in the way you will meet it.