The normal distribution is a symmetric, bell-shaped curve that describes many naturally occurring data sets, like heights or test scores, where most values cluster around the mean and fewer values occur further away. The empirical rule (or 68-95-99.7 rule) states that about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three — standard deviation measures how spread out the data is.
Example
If a class's test scores are normally distributed with a mean of 70 and a standard deviation of 10, about 68% of students scored between 60 and 80, and about 95% scored between 50 and 90 — a student scoring 95 would be unusually high, well outside two standard deviations.
Key terms
Normal distribution:
A symmetric, bell-shaped distribution where data clusters around the mean.
Standard deviation:
A measure of how spread out a data set is from its mean.
68-95-99.7 rule:
The proportions of data within one, two and three standard deviations of the mean.
Questions
1. A normal distribution is shaped like:
A symmetric bell curve
A perfectly straight line
A random scatter with no pattern
A single point
2. In a normal distribution, most values cluster:
Around the mean
Only at the very extremes
Nowhere in particular
Only below zero
3. Standard deviation measures:
How spread out a data set is
The exact middle value only
The total number of data points
The highest single value only
4. According to the empirical rule, about 68% of values fall within:
One standard deviation of the mean
Ten standard deviations of the mean
Zero standard deviations of the mean
Only the exact mean value
5. According to the empirical rule, about 95% of values fall within:
Two standard deviations of the mean
Half a standard deviation of the mean
Ten standard deviations of the mean
Only the exact mean value
6. A smaller standard deviation means the data is:
More tightly clustered around the mean
More spread out from the mean
Impossible to graph
Always exactly normal
7. A normal distribution curve is:
Symmetric
Always lopsided to one side
A straight diagonal line
Made of disconnected points only
8. If a class has a mean test score of 70 and standard deviation of 10, about 68% of scores fall between:
60 and 80
50 and 90
70 and 100
0 and 70
9. Using the same class (mean 70, standard deviation 10), about 95% of scores fall between:
50 and 90
60 and 80
40 and 100
65 and 75
10. A student scoring 95 in a class with mean 70 and standard deviation 10 is:
More than two standard deviations above the mean
Exactly at the mean
Below the mean
Within one standard deviation of the mean
11. A score exactly at the mean in a normal distribution is:
The most common or typical score
The rarest possible score
Always impossible to achieve
Always the lowest score
12. Two data sets with the same mean but different standard deviations will have:
Different amounts of spread around that mean
The exact same shape and spread
No relationship to each other at all
Identical minimum and maximum values
13. Why is the normal distribution often used to model naturally occurring data like heights, rather than assuming values are spread completely randomly?
Many natural traits cluster around a typical value with fewer extreme cases, matching the bell-curve pattern
Natural data is always spread in a completely flat, even pattern
The normal distribution has no real connection to naturally occurring data
Height and similar traits are always distributed in a perfectly straight line
14. Why might understanding standard deviation help interpret whether a specific data point is unusual, not just the mean alone?
The mean alone doesn't show how typical or extreme a value is relative to the overall spread
The mean always provides complete information with no need for standard deviation
Standard deviation has no connection to identifying unusual values
Every value is equally typical regardless of standard deviation
15. Why is the 68-95-99.7 rule useful for quickly estimating probabilities without complex calculations?
It gives fixed, memorable proportions for how data clusters at set distances from the mean in any normal distribution
This rule only applies to a single specific data set and never generalises
The rule requires more complex calculation than working out probabilities from scratch
The percentages in this rule change randomly for every different data set
16. Why might comparing an individual's test score to the class mean and standard deviation give more insight than comparing it to the mean alone?
Standard deviation shows how significant the difference from the mean actually is, given the overall spread
Standard deviation is completely irrelevant when interpreting an individual score
The mean alone always provides the full picture, with no need for further context
Comparing to the mean and standard deviation together always gives less insight than the mean alone
17. Why might real-world data only approximate a normal distribution rather than being perfectly bell-shaped?
Real data is subject to outliers, sample size effects and non-random influences that create some deviation from a perfect curve
Real-world data is always mathematically identical to a perfect normal distribution
Sample size and outliers have no effect on the shape of a data set
A perfect bell curve always exactly matches every real-world data set
18. If a class has a mean test score of 65 and standard deviation of 8, a score of 73 is:
One standard deviation above the mean
Two standard deviations above the mean
Below the mean
Exactly at the mean
19. Which data set is more tightly clustered around its mean?
A data set with standard deviation 2
A data set with standard deviation 15
Both are equally clustered, always
Standard deviation cannot indicate clustering
20. Why might manufacturers use the normal distribution and standard deviation to set acceptable tolerance ranges for a product's dimensions?
It lets them predict what proportion of products will fall within an acceptable range around the target measurement
Manufacturing measurements are never modelled using statistical distributions
Standard deviation has no practical application in quality control
All manufactured products are always identical with no variation at all
21. Why might a value more than three standard deviations from the mean be considered a significant outlier in a normal distribution?
Only about 0.3% of values fall beyond three standard deviations, making such a value extremely rare under a normal model
Values beyond three standard deviations are actually the most common in a normal distribution
The empirical rule does not apply beyond two standard deviations
Outliers are defined only by how large a raw number is, not its position relative to the mean
Answer key (parent copy)
1. A symmetric bell curve
2. Around the mean
3. How spread out a data set is
4. One standard deviation of the mean
5. Two standard deviations of the mean
6. More tightly clustered around the mean
7. Symmetric
8. 60 and 80
9. 50 and 90
10. More than two standard deviations above the mean
11. The most common or typical score
12. Different amounts of spread around that mean
13. Many natural traits cluster around a typical value with fewer extreme cases, matching the bell-curve pattern
14. The mean alone doesn't show how typical or extreme a value is relative to the overall spread
15. It gives fixed, memorable proportions for how data clusters at set distances from the mean in any normal distribution
16. Standard deviation shows how significant the difference from the mean actually is, given the overall spread
17. Real data is subject to outliers, sample size effects and non-random influences that create some deviation from a perfect curve
18. One standard deviation above the mean
19. A data set with standard deviation 2
20. It lets them predict what proportion of products will fall within an acceptable range around the target measurement
21. Only about 0.3% of values fall beyond three standard deviations, making such a value extremely rare under a normal model