This cheat sheet covers how to use z-scores and the standard normal distribution to compare data values from different normal distributions. Students need these tools to find probabilities, percentiles, and unusual values in statistics problems. A strong understanding of z-scores also helps with sampling distributions and later inference topics.
The reference gives a quick way to connect raw scores, standardized scores, and table or calculator probabilities.
The main formula is , where is a data value, is the mean, and is the standard deviation. A positive z-score is above the mean, a negative z-score is below the mean, and is exactly at the mean. The standard normal distribution has mean and standard deviation .
Areas under the normal curve represent probabilities, so is found by subtracting cumulative areas.
Key Facts
- The z-score formula is for a population mean and population standard deviation .
- To convert a z-score back to a raw value, use .
- The standard normal random variable has mean and standard deviation .
- A z-score tells how many standard deviations a value is from the mean, so means standard deviations above the mean.
- For cumulative probability, is the area to the left of under the standard normal curve.
- To find an interval probability, use .
- By symmetry of the standard normal curve, .
- The empirical rule says about of normal data are within , within , and within of the mean.
Vocabulary
- Z-score
- A z-score is a standardized value that tells how many standard deviations a data value is from the mean.
- Standard normal distribution
- The standard normal distribution is a normal distribution with mean and standard deviation .
- Cumulative probability
- Cumulative probability is the area under a distribution curve to the left of a given value, written as .
- Percentile
- A percentile is the percentage of values in a distribution that are at or below a given value.
- Normal curve
- A normal curve is a symmetric, bell-shaped curve used to model many real data distributions.
- Tail area
- A tail area is the probability in the far left or far right end of a distribution beyond a selected cutoff.
Common Mistakes to Avoid
- Using instead of is wrong because it reverses the sign and changes whether the value is above or below the mean.
- Forgetting to subtract cumulative areas for an interval is wrong because is not the same as .
- Treating a z-score as a probability is wrong because a z-score is a location, while probability is an area under the curve.
- Using the standard normal table without checking whether it gives left-tail area is wrong because some tables report different area types.
- Rounding z-scores too early is wrong because small rounding changes can noticeably affect probabilities and percentiles.
Practice Questions
- 1 A test has mean and standard deviation . Find the z-score for a student who scored .
- 2 A normal distribution has and . What raw value corresponds to ?
- 3 Using a standard normal table or calculator, find .
- 4 A value has in one class and another value has in another class. Explain how their locations compare relative to their own class distributions.
Understanding Z-Score & Standard Normal Reference
A z-score is useful because raw numbers can hide the context of a measurement. A score of eighty may be excellent on one test and ordinary on another. The mean tells what is typical for that group.
The standard deviation tells how spread out the group is. Standardizing combines both pieces of information into one position on a common scale. This makes it possible to compare a student’s test result with a runner’s race time, provided each value is judged within its own distribution.
The direction matters for interpretation. A high time is usually worse in a race, even though it has a positive z-score when it is above the mean.
Normal tables and many calculators usually report cumulative area, meaning the proportion below a chosen z-score. This is not always the probability requested in a problem. A phrase such as greater than means use the area to the right, which is one minus the reported left area.
A phrase such as between means find the left area at each endpoint, then subtract the smaller area from the larger one. A phrase such as at least or no more than includes an endpoint, but this makes no numerical difference for a continuous normal model. A single exact value has probability zero because the curve represents a continuum of possible values, not separate bars.
Percentile problems work in the reverse direction. First identify the cumulative proportion below the desired percentile. Then use a table or an inverse normal calculator to find the z-score with that left area.
Finally, convert that standardized location into a measurement using the mean and standard deviation of the situation. For example, the ninetieth percentile is not ninety points or ninety percent correct by itself. It is the value above which about ten percent of observations fall.
Schools use percentiles in standardized testing. Health charts use them to compare growth with a reference population. Quality control uses them to set limits for manufactured parts.
The normal model is an approximation, not a rule that every data set obeys. Before using z-score probabilities, inspect the shape of the data or read the problem carefully for a normality assumption. A distribution with strong skew, two separate clusters, or extreme outliers may give misleading normal estimates.
The empirical rule is a quick reasonableness check. Values more than two standard deviations from the mean deserve attention, while values beyond three standard deviations are rare in a well fitting normal distribution. Rare does not mean impossible or automatically wrong.
Students should track whether a question asks for a raw value, a standardized score, a left-tail area, a right-tail area, or a middle area. Most mistakes come from choosing the wrong region of the curve or reversing a subtraction.