Intro to Z-Scores
This section introduces you to z-scores. When I think of statistics, one of the first things that come in my mind is standard deviation and z-scores. So, what exactly are z-scores? A z-score, also known as a standard score, is a measure of how many standard deviations a data point is from the mean (not median) of a data set. It is calculated by subtracting the mean of the data set from the value of the data point, and then dividing the result by the standard deviation of the data set. The formula is simple but very powerful. It is resistant to units, and it can be used to compare any activity.
FORMULA: z = (x - x̄) / s
where z = z-score, x = a data point, x̄ = mean value, s = standard deviation
For example, consider a data set with a mean of 50 and a standard deviation of 10. If a data point has a value of 70, the z-score for that data point would be calculated as follows:
z-score = (70 - 50) / 10 = 2
This z-score of 2 means that the data point is 2 standard deviations above the mean of the data set.
z-scores are useful for comparing values within a data set and for determining whether a value is unusual or extreme relative to the rest of the data. They can also be used to standardize data for comparison between different data sets.
For this reason, z-scores are also called standardized values. In sports, when the judges have to calculate the final score for athletes, they use z-scores.
Negative z-scores mean that the data value is below the mean, while positive z-scores mean that the data value is higher than the mean. The further the value is from the mean, irrespective of the sign, the more unusual the value is. Here is the formula for z-score:
As you see, when we are standardizing data into z-scores, we are shifting them by the mean and rescaling by the standard deviation.
Wait, but how does standardization affect the distribution?
In general, shifting data changes the distribution but leaves the shape and spread unchanged. The center shifts with other measures of the position such as percentiles, mininum, and maximum by the same amount of value.
What about rescaling? You may guess already that with rescaling data when we multiply or divide any number to a data set, the shape of distribution won’t change (it will just look stretched or squeezed), but everything else will change, the mean, minimum, maximum, range, IQR, and standard deviation. AP Statistics MCQs always will ask questions like this to trick you if you know how the shifting and rescaling affect the shape, center, and spread, so get ready to encounter such questions!
Normal Model: More than Just a Hump
You may have learned about "normal" models or bell-shaped curves in your Algebra class and through calculus. Some sets of data may be described as approximately normally distributed. A normal curve is mound-shaped and symmetric.
Normal models are appropriate for symmetric and unimodal distributions. The normal model has two parameters (the population mean, µ, and the population standard deviation, σ) and is often written as N(mean, sd). These parameters do not come from data but are part of the model.
You might be curious: what variables in daily life can a normal distribution model. The answer? A lot! Here's a list of a small sample (get it?) of all the variables out there that follow a normal distribution:
- Height
- IQ scores
- Blood pressure
- Birth weight
- Body temperature
- Life expectancy
- Income
Standard Normal Model
The standard normal model has a symmetrical bell-shaped curve, with the mean at 0 and the standard deviation at 1. Z-scores are actually based on the standard normal model. When working with data that is normally distributed, it is often helpful to standardize the data by converting the values to z-scores, which allows for easier comparison and analysis. This standard model can be written as N(0,1), and to standardize, we need to subtract from mean and rescale by the standard deviation. z = (x - x̄) / s
The standard normal model, as well as other normal distributions, are based on the assumption that the data follows a symmetrical, bell-shaped curve. In order for the standard normal model or other normal distributions to be a good model for the data, the data must be approximately symmetric and unimodal.
If the data is not symmetric or is multimodal (i.e. has multiple peaks), then the standard normal model or other normal distributions may not be a good fit for the data. In such cases, it may be necessary to use a different statistical model or transform the data in some way to make it more suitable for analysis.
To check whether the data is approximately symmetric and unimodal, it is common to look at the histogram of the data or create a normal probability plot. A histogram should show a roughly symmetrical distribution, with a single peak in the middle. Don’t model data with a Normal model without checking the "Nearly Normal" Condition.
The Empirical (68–95–99.7) Rule
Often we ask ourselves whether we are normal or not. If we are normal, then we should be doing about the same things as the average people do. The 68-95-99.7 rule (Empirical Rule) tells us that if we all behave normally then about 68% of the values fall within one standard deviation of the mean, about 95% of the values fall within two standard deviations of the mean, and about 99.7%—almost all—of the values fall within three standard deviations of the mean.
The EMPIRICAL RULE: "For a normal distribution, approximately 68% of the observations are within 1 standard deviation of the mean, approximately 95% of observations are within 2 standard deviations of the mean, and approximately 99.7% of observations are within 3 standard deviations of the mean."
This rule works fine in normal models, but do not ever try it for skewed distributions as it will fail. For skewed distributions, we can use Chebisheeve’s (a Russian mathematician) rule, but that’s beyond the AP Statistics course. When sketching the normal model, start with the center and extend the tails to the sides, but you do not need to go beyond three standard deviations as there is very little left beyond it, and also don’t touch the line because it extends forever. The place where the bell shape starts to curve downward is called the inflection point, which is exactly one standard deviation away from the mean.
🎥 Watch: AP Stats - Normal Distributions
🎥 Watch: AP Stats - Normal Curve and Normal Calculations
Key Vocabulary
- Density Curve
- Normal Distribution
- Normal Curve
- The Empirical Rule
- Standard Normal Distribution
- Z-Score
- Normal Probability Plot
Practice Problems
(1) A sample of 50 students at a school took a math test, and the mean score was 75 out of 100. The standard deviation of the scores was 10.
A. Calculate the z-score for a student who scored a 90 on the test.
B. Interpret the z-score. What does this mean in the context of statistics and test scores?
(2) A baseball player has a batting average of 0.300, which is the mean number of hits per at-bat over the course of a season. The standard deviation of the player's batting average is 0.050. In a recent game, the player had 4 at-bats and scored 3 hits. Calculate the z-score for the player's performance in this game.
(3) Another big idea within Unit 1 of the AP Stats course is the idea that percentiles and z-scores may be used to compare relative positions of points within a data set or between data sets.
A study was conducted to determine the average number of hours per week that college students spend studying. The study found that the average number of hours per week spent studying is 15 hours, with a standard deviation of 4 hours. A random sample of 25 college students was selected and the number of hours they spent studying per week was recorded. The sample mean was found to be 13 hours per week, with a z-score of -1.5.
Based on the information provided, what is the average number of hours per week that college students spend studying?
Answers
(1) A. First, we subtract the mean score from the student's score to find the difference:
90 - 75 = 15
Next, we divide the difference by the standard deviation of the scores:
15 / 10 = 1.5
B. The z-score for the student's score is 1.5, which means that the student scored 1.5 standard deviations above the mean test score.
Note: When calculating z-scores, it is important to use the same units for the mean, standard deviation, and data point being analyzed. In this example, all values are in units of points on the test. If the mean and standard deviation were in different units (e.g. percent instead of points), it would be necessary to convert the values to the same units before calculating the z-score.
(2) First, we calculate the player's batting average for the game by dividing the number of hits by the number of at-bats:
batting average = 3 / 4 = 0.750
Next, we subtract the mean batting average from the game batting average to find the difference:
0.750 - 0.300 = 0.450
Finally, we divide the difference by the standard deviation of the player's batting average:
0.450 / 0.050 = 9
The z-score for the player's performance in this game is 9, which means that the player's performance was 9 standard deviations above the mean. This indicates that the player had a very strong performance in this game!
(3) To solve this problem, we can use the formula for calculating a z-score:
z = (x - x̄) / standard deviation
In this case, the mean is 15 hours per week and the standard deviation is 4 hours per week. The value we are trying to find, the average number of hours per week that college students spend studying, is represented by x.
Substituting these values into the formula, we get:
z = (x - 15) / 4
We are given that the z-score is -1.5, so substituting this value into the formula gives us:
-1.5 = (x - 15) / 4
Multiplying both sides of the equation by 4 gives us:
-6 = x - 15
Adding 15 to both sides of the equation gives us:
x = 9
Therefore, the average number of hours per week that college students spend studying is 9 hours! Does that sound reasonable in your experience? Hmm...
Vocabulary
The following words are mentioned explicitly in the College Board Course and Exam Description for this topic.
| Term | Definition |
|---|---|
| approximately normally distributed | A description of data sets that closely follow the pattern of a normal distribution with a mound-shaped, symmetric curve. |
| empirical rule | A rule stating that for a normal distribution, approximately 68% of observations fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. |
| mean | The average value of a dataset, represented by μ in the context of a population. |
| normal curve | The bell-shaped graph of a normal distribution that is symmetric and mound-shaped. |
| normal distribution | A probability distribution that is mound-shaped and symmetric, characterized by a population mean (μ) and population standard deviation (σ). |
| normally distributed random variable | A random variable that follows a normal distribution, allowing for the calculation of probabilities for specific intervals. |
| parameter | A numerical summary that describes a characteristic of an entire population. |
| percentile | A value such that p% of the data is less than or equal to it, used to describe the position of a data point within a distribution. |
| population mean | The average of all values in an entire population, denoted as μ. |
| population means | The average values of two distinct populations being compared, denoted as μ₁ and μ₂. |
| population standard deviation | A measure of the spread or dispersion of all values in a population, denoted by σ, which is a parameter of the normal distribution. |
| proportion | A part or share of a whole, expressed as a fraction, decimal, or percentage. |
| relative position | The location of a data point within a data set, often expressed in comparison to other values or as a measure of how it ranks relative to the distribution. |
| standard deviation | A measure of how spread out data values are from the mean, represented by σ in the context of a population. |
| standard normal table | A reference table that provides the cumulative probabilities (areas under the curve) for the standard normal distribution. |
| z-score | A standardized score calculated as (xi - μ)/σ that measures how many standard deviations a data value is from the mean. |
Frequently Asked Questions
How do I calculate a z-score and what does it actually mean?
A z-score tells you how many standard deviations a data value is from the mean. You calculate it with z = (x − μ) / σ for a population (or z = (x − x̄) / s for a sample when you’re standardizing). Example: if μ = 100, σ = 15 and x = 130, z = (130−100)/15 = 2, so x is 2 standard deviations above the mean. Why it matters: z-scores let you compare values from different distributions, find percentiles with the standard normal (z) table or calculator, and use the empirical rule (≈68% within 1σ, 95% within 2σ, 99.7% within 3σ) to judge how unusual a value is. On the AP exam you can use a calculator or z-table (tables are provided), so standardize first, then find area/probability with technology or the table. For more review, see the Topic 1.10 study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and practice problems (https://library.fiveable.me/practice/ap-statistics).
What's the formula for finding percentiles in a normal distribution?
Use the z-score formula and its inverse. - To convert a data value x to a z-score: z = (x − μ) / σ. - To find the x that corresponds to a given percentile p: first find the standard normal z* with area p below it (using a z-table or calculator), then use x = μ + z*·σ. Example: the 90th percentile has z* ≈ 1.28, so x90 = μ + 1.28σ. This is exactly what VAR-2.B in the CED expects: standardize with z-scores, use tables/technology to get areas, and convert back. On the AP exam you may use the provided z-table or your calculator (formula sheet/tables are given). For a quick refresher and practice, see the Topic 1.10 study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and more practice problems (https://library.fiveable.me/practice/ap-statistics).
When do I use the empirical rule vs calculating exact probabilities?
Use the empirical rule when you need a quick, approximate idea and the distribution is approximately normal (mound-shaped, symmetric). The 68–95–99.7 rule (VAR-2.A.3) tells you roughly what percent of observations lie within 1, 2, and 3 σ of μ—great for fast estimates or checking whether data look plausibly normal. Do exact probabilities when you need precision or the interval isn’t exactly ±1, ±2, ±3 σ. Standardize with a z-score (z = (x−μ)/σ, VAR-2.B.2) and use a z-table, calculator, or software (VAR-2.B.3) to get exact areas (tails, percentiles, or arbitrary intervals). Also use exact methods if the distribution isn’t clearly normal or the question is graded for precise probability (AP free-response/multiple choice expect correct areas). For AP prep, practice both: use the study guide for Topic 1.10 (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and drill exact-area problems with the practice set (https://library.fiveable.me/practice/ap-statistics).
What's the difference between a parameter and a statistic in normal distributions?
A parameter is a number that describes a whole population—for a normal model those are the population mean μ and population standard deviation σ (CED VAR-2.A.2). A statistic is a number computed from a sample: common ones are the sample mean x̄ and the sample standard deviation s. You use statistics to estimate parameters (e.g., x̄ estimates μ, s estimates σ) and to carry out inference. In normal-distribution problems you’ll often standardize values using z = (x − μ)/σ when you know the population parameters, or use t procedures (with s) when you only have sample statistics. For AP exam focus: know the difference in wording—“parameter” refers to population μ or σ; “statistic” refers to sample x̄ or s—and how sampling distributions let you make probability statements about statistics. For a quick refresher, see the Topic 1.10 study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and practice problems (https://library.fiveable.me/practice/ap-statistics).
I'm confused about how to tell if my data is normally distributed - what should I look for?
Look for shape, center, spread, and outliers. A roughly normal distribution is mound-shaped and symmetric (mean ≈ median), with no strong skew or big outliers. Use these checks: - Graphical: draw a histogram or a normal (QQ) plot / normal probability plot. On a QQ-plot, points should lie close to a straight line. - Numeric: compare mean and median; compute z-scores to see if extreme values are plausible. - Empirical rule: about 68% of observations should fall within ±1σ of μ, ~95% within ±2σ, and ~99.7% within ±3σ. Big deviations from these percentages suggest nonnormality. - Sample size: small samples make visual checks noisy; large samples reveal departures more clearly. For AP problems, justify your conclusion by citing shape and the empirical rule or a QQ-plot. If you need a quick refresher, see the Topic 1.10 study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and try practice questions (https://library.fiveable.me/practice/ap-statistics).
How do I solve problems where I need to find the area under a normal curve?
You follow three steps every time: 1. Standardize. Convert x to a z-score: z = (x − μ) / σ. This tells you how many standard deviations x is from the mean (CED VAR-2.B.1–2). 2. Decide the area you want: - P(X > a): area to the right of z. - P(X < a): area to the left of z. - P(b < X < a): area between two z’s. 3. Use technology or a z-table to get the area (CED VAR-2.B.3). On the AP exam you may use a graphing calculator: normalcdf(lower, upper, μ, σ) or normalcdf(lower z, upper z, 0, 1) for Z. For example, if μ = 100, σ = 15 and you want P(X > 120): z = (120−100)/15 = 1.33. Find the area below z (≈0.908), so P(X > 120) ≈ 1 − 0.908 = 0.092. Use the empirical rule (68–95–99.7) for quick estimates (CED VAR-2.A.3). For more examples and practice, see the Unit 1 normal distribution study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and try practice problems (https://library.fiveable.me/practice/ap-statistics).
Can someone explain the 68-95-99.7 rule step by step with an example?
Think of a normal distribution as a symmetric bell with mean μ and standard deviation σ. The empirical rule (68–95–99.7) tells you roughly how much of the data lies within 1, 2, and 3 standard deviations of the mean. Step-by-step with an example: - Suppose test scores are approximately normal with μ = 80 and σ = 6. - Within 1σ: 80 ± 6 → 74 to 86. About 68% of students score between 74 and 86. - Within 2σ: 80 ± 2·6 → 68 to 92. About 95% score between 68 and 92. - Within 3σ: 80 ± 3·6 → 62 to 98. About 99.7% score between 62 and 98. Use z-scores to standardize: z = (x − μ)/σ. For x = 92, z = (92−80)/6 = 2 → about the 97.5th percentile (right tail beyond +2σ is ~2.5%). For AP Stats, you should know this rule (VAR-2.A.3) and how to standardize and find areas with tables or technology (VAR-2.B). Want more examples and practice? Check the Topic 1.10 study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and try problems at (https://library.fiveable.me/practice/ap-statistics).
What's the difference between using a z-table and using my calculator for normal distribution problems?
Short answer: both find areas under the standard normal curve, but they work differently and each has pros. Z-table (Table A) - Gives Φ(z) = area to the left of a z-value from the standard normal. - You’ll need to standardize x → z = (x − μ)/σ first (VAR-2.B.1–B.2). - Use symmetry and subtraction for right-tail or two-sided areas; limited to the grid values (you may interpolate). - Useful for understanding the empirical rule (68–95–99.7) and doing hand checks. Calculator / technology - Can compute any area or percentile directly (normalcdf, invNorm, etc.), for any μ and σ (VAR-2.B.3–B.4). - Faster, more precise, handles tails and inverse problems without manual complements. - AP permits a graphing calculator on the exam and tables are also provided—so either is allowed, but using a calculator is quicker for messy numbers. If you want practice switching between table and tech, check the Topic 1.10 study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and try problems at (https://library.fiveable.me/practice/ap-statistics).
How do I know when a variable can be modeled by a normal distribution?
You can model a variable with a normal distribution when the population (or the sampling distribution) is roughly mound-shaped and symmetric with no strong skew or outliers. The normal model is fully described by the population mean μ and standard deviation σ, and follows the empirical rule: about 68% within ±1σ, 95% within ±2σ, and 99.7% within ±3σ. Check your data with a histogram or normal probability plot and compute z-scores to see if values fall where the normal curve predicts. For sample means, the Central Limit Theorem says the sampling distribution is approximately normal for moderate-to-large n even if the original data aren’t normal—so sample size matters. On the AP exam you’ll be asked to compare a distribution to the normal model (CED VAR-2.A, VAR-2.B); use graphs, the empirical rule, and z-scores, and rely on technology when allowed. For a concise review, see the Topic 1.10 study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8). For broader unit review or practice problems, use (https://library.fiveable.me/ap-statistics/unit-1) and (https://library.fiveable.me/practice/ap-statistics).
I don't understand how to compare data points from different normal distributions using z-scores.
Think of a z-score as “how many standard deviations away from the mean” a value is. To compare values from different normal distributions, convert each value x to a z-score: z = (x − μ)/σ. Once you standardize both values, they’re on the same scale (the standard normal), so larger z means more extreme above the mean, smaller (more negative) z means more extreme below. Example: a 78 on Test A with μ=70, σ=5 gives z=(78−70)/5=1.6. A 92 on Test B with μ=86, σ=3 gives z=(92−86)/3=2.0—the 92 is relatively farther above its mean than the 78 is above its mean. Use the empirical rule (68–95–99.7) or a z-table/technology to get percentiles/proportions from z (CED VAR-2.B, VAR-2.C). For more practice and AP-aligned notes on normal models and z-scores, see the Topic 1.10 study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) or the Unit 1 overview (https://library.fiveable.me/ap-statistics/unit-1). For lots of practice problems, try (https://library.fiveable.me/practice/ap-statistics).
How do I find what percentage of data falls between two values in a normal distribution?
You do three things: standardize, find areas, subtract. 1. Standardize each value x to a z-score: z = (x − μ)/σ. That tells you how many σ’s each x is from the mean (CED VAR-2.B.1–B.2). 2. Use a standard normal table or technology to get the area (probability) below each z: Φ(z1) and Φ(z2). On the AP exam you may use a calculator or table (CED VAR-2.B.3). 3. The percent between the two x’s = [Φ(z2) − Φ(z1)] × 100%. Quick example: μ = 100, σ = 15, want percent between 85 and 118. z1 = (85−100)/15 = −1.00, z2 = (118−100)/15 = 1.20. Look up Φ(1.20) ≈ 0.8849 and Φ(−1.00) ≈ 0.1587, so area ≈ 0.8849−0.1587 = 0.7262 → 72.62%. If you need shortcuts, use the empirical rule (68–95–99.7) for rough estimates. For step-by-step practice, see the Normal Distribution study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and more practice problems (https://library.fiveable.me/practice/ap-statistics).
What does it mean when a z-score is negative vs positive?
A z-score = (x − μ)/σ tells you how many standard deviations a value is from the mean. If z is positive, the data value is above the mean; if z is negative, it’s below the mean. The sign gives direction (above/below) and the magnitude gives distance in standard deviations—e.g., z = 2 means 2 SDs above the mean, z = −1.5 means 1.5 SDs below. For a normal distribution you can use the empirical rule: about 68% of values fall within ±1 SD, 95% within ±2 SD, and 99.7% within ±3 SD—so large |z| values are rare. To find proportions or percentiles use the standard normal (z) table or your calculator (AP exam allows z-tables/technology). This ties directly to VAR-2.B in the CED, which defines z-scores and using technology to find areas. If you want a quick refresher, check the Topic 1.10 normal distribution study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and try practice problems (https://library.fiveable.me/practice/ap-statistics).
How do I work backwards from a percentile to find the actual data value?
To go from a percentile to an actual data value for a normal model, do two steps: 1. Find the z-score for that percentile (use a standard normal table or your calculator’s inverse normal function). For example, the 90th percentile corresponds to z ≈ 1.28 (invNorm(0.90)). 2. Convert that z to the original units with x = μ + zσ (CED VAR-2.B.1–B.2). So if μ = 50 and σ = 8 and you want the 90th percentile: x = 50 + 1.28(8) ≈ 60.2. Notes: the z formula is z = (x − μ)/σ. On the AP exam you may use technology (invNorm on your graphing calculator) as allowed; just state the method and show x = μ + zσ if asked (CED VAR-2.B.3). For extra practice and worked examples, see the Normal Distribution study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and more problems at (https://library.fiveable.me/practice/ap-statistics).
Why is the normal distribution always symmetric and bell-shaped?
Because the normal is defined so height depends only on how far a value is from the mean, it’s always symmetric and “mound-shaped.” Mathematically the normal pdf uses (x − μ)² in the exponent, so f(μ + d) = f(μ − d): distances equal on either side of μ give equal probability, which makes it symmetric. It’s bell-shaped because the density is highest at μ and decreases smoothly as |x − μ| grows; the rate of decrease is set by σ (larger σ = wider, flatter bell). That shape gives the empirical rule: ≈68%, 95%, 99.7% within 1, 2, 3 σ of μ (CED VAR-2.A.2–3). On the AP exam you’ll compare data to this model and use z-scores and tables/technology to find proportions (VAR-2.B). For a quick refresher, see the Topic 1.10 study guide (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and try practice problems (https://library.fiveable.me/practice/ap-statistics).
I keep getting confused about when to use mu and sigma vs x-bar and s - can someone explain?
Short answer: use μ and σ when you’re talking about the population (the true distribution, a parameter). Use x̄ and s when you’re talking about a sample (what you actually measured). Why it matters: the CED calls μ and σ the parameters of a normal distribution (VAR-2.A.2). If the population mean and SD are known, z-scores use x and (x−μ)/σ (VAR-2.B). But in real life you usually only have a sample, so you estimate the center and spread with x̄ and s; those feed into sample-based statistics and standard errors (like s/√n) and t-based inference when σ is unknown (see sampling distributions in the formula sheet). On the exam: label things correctly—if the question asks about a population or model, use μ, σ; for sample results or confidence intervals use x̄, s (and t critical values if σ unknown). For a quick review, check the Normal Distribution study guide on Fiveable (https://library.fiveable.me/ap-statistics/unit-1/normal-distribution/study-guide/f842Kr6YNnYX4G0dtAC8) and try practice problems (https://library.fiveable.me/practice/ap-statistics).