🎲Intro to Statistics Unit 6 Review

Z-scores and the standard normal distribution let you take any normally distributed data and convert it to a common scale. This makes it possible to compare values across completely different datasets, like test scores from two different classes with different averages. The Empirical Rule then gives you a fast way to estimate how much data falls within certain ranges, no calculator or table required.

Z-Scores

A z-score tells you how many standard deviations a particular value sits from the mean. The formula is:

$z = \frac{x - \mu}{\sigma}$

$x$ = the value you're looking at
$\mu$ = the population mean
$\sigma$ = the population standard deviation

How to interpret z-scores:

A positive z-score means the value is above the mean.
A negative z-score means the value is below the mean.
A z-score of 0 means the value equals the mean exactly.

The absolute value of the z-score tells you how far from the mean. For example, a z-score of -2.5 means the value is 2.5 standard deviations below the mean, while a z-score of 1.8 means 1.8 standard deviations above.

Quick example: Suppose exam scores have a mean of 75 and a standard deviation of 10. If you scored an 90:

$z = \frac{90 - 75}{10} = 1.5$

Your score is 1.5 standard deviations above the mean. Z-scores like this can also be used to find percentiles, which tell you what percentage of values fall below yours.

The Empirical Rule (68-95-99.7 Rule)

The Empirical Rule applies to normal distributions and gives you a quick estimate of how data spreads around the mean:

68% of data falls within ±1 standard deviation of the mean
95% of data falls within ±2 standard deviations of the mean
99.7% of data falls within ±3 standard deviations of the mean

You can also use these percentages to figure out the data in between specific boundaries. Since the normal distribution is symmetric, each half of a range contains an equal share. For instance:

From the mean to +1 standard deviation contains about 34% of the data (half of 68%).
From +1 to +2 standard deviations contains about 13.5% (half of 95% minus half of 68%).

So the percentage of data between -1 and +2 standard deviations is 34% + 34% + 13.5% = 81.5%. Drawing a quick sketch of the bell curve and labeling these percentages makes problems like this much easier to work through.

Transforming to the Standard Normal Distribution

Any normal distribution can be converted to the standard normal distribution, which has a mean of 0 and a standard deviation of 1. Here's how:

Start with a value $x$ from a normal distribution with mean $\mu$ and standard deviation $\sigma$ .
Calculate the z-score: $z = \frac{x - \mu}{\sigma}$
The resulting z-score is now a value on the standard normal distribution.

The standard normal distribution has two key properties:

The total area under the curve equals 1 (representing 100% of the data).
The curve is perfectly symmetric about $z = 0$ .

Why bother standardizing? Because it lets you compare values from different distributions on equal footing. Say one class has a mean of 70 with a standard deviation of 8, and another has a mean of 85 with a standard deviation of 5. A raw score of 78 in the first class and 90 in the second class look different, but converting both to z-scores (both equal 1.0) shows they represent the same relative position.

Probability and the Standard Normal Distribution

Once you've converted to z-scores, you can use the standard normal table (or a calculator) to find probabilities. The table gives you the area under the curve to the left of a given z-score, which represents the probability of a value falling at or below that point. This area is called the cumulative distribution function (CDF).

For example, a z-score of 1.5 corresponds to a CDF value of about 0.9332, meaning roughly 93.3% of data falls below that value.

These probabilities become the foundation for later topics like confidence intervals, where you'll use z-scores to estimate population parameters within a specified level of confidence.

🎲Intro to Statistics Unit 6 Review