6.1 The Standard Normal Distribution

The Standard Normal Distribution

The standard normal distribution gives you a way to compare data points that come from completely different scales. By converting raw values into z-scores, you can place any normally distributed data on the same scale and directly compare them.

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Calculation of Z-Scores

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

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

• $x$ = the individual data point
• $\mu$ = the mean of the distribution
• $\sigma$ = the standard deviation of the distribution

The sign of the z-score tells you which side of the mean you're on. A positive z-score means the value is above the mean, and a negative z-score means it's below the mean. A z-score of 0 means the value is the mean.

The real power of z-scores is that they let you compare values from distributions with different units and scales. For example, suppose you score 1200 on the SAT (mean 1060, SD 200) and 28 on the ACT (mean 21, SD 5). Which performance was stronger?

1. SAT z-score: $z = \frac{1200 - 1060}{200} = 0.70$

2. ACT z-score: $z = \frac{28 - 21}{5} = 1.40$

The ACT z-score is higher, so relative to other test-takers, you performed better on the ACT. Once standardized, every z-score distribution has a mean of 0 and a standard deviation of 1, which is what makes this comparison possible.

Interpretation of Z-Scores

The standard normal distribution (also called the z-distribution) has a mean of 0 and a standard deviation of 1. Every z-score maps to a specific position on this distribution, and the area under the curve tells you probabilities.

Some useful reference points:

• A z-score of 0 corresponds to the 50th percentile (the mean)
• A z-score of 1 corresponds to roughly the 84th percentile
• A z-score of -1 corresponds to roughly the 16th percentile
• A z-score of 2 corresponds to roughly the 97.5th percentile

The probability of a data point falling within a certain range equals the area under the curve between those two z-scores. For instance, the area between $z = -1$ and $z = 1$ is about 0.68, meaning there's a 68% chance a randomly selected value falls within one standard deviation of the mean.

The probability density function describes the height of the curve at each point. For continuous distributions, you don't find the probability of a single exact value; instead, you find the probability over an interval (an area under the curve).

Empirical Rule for Z-Score Probabilities

The empirical rule (also called the 68-95-99.7 rule) gives you a quick way to estimate probabilities without a table or calculator. For any normal distribution:

• ~68% of data falls within $\mu \pm 1\sigma$ (z-scores between -1 and 1)
• ~95% of data falls within $\mu \pm 2\sigma$ (z-scores between -2 and 2)
• ~99.7% of data falls within $\mu \pm 3\sigma$ (z-scores between -3 and 3)

To apply the empirical rule:

1. Convert the given values to z-scores using $z = \frac{x - \mu}{\sigma}$

2. Determine how many standard deviations those z-scores are from the mean

3. Apply the matching percentage from the rule

You can also use these percentages to find probabilities in the tails. Since 95% of data falls between $z = -2$ and $z = 2$, the remaining 5% is split between the two tails, so about 2.5% falls above $z = 2$.

The empirical rule is great for estimation, but for precise probabilities (like the area between $z = 0.5$ and $z = 1.3$), you'll need a z-table or calculator.

Additional Concepts

The cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a specific value. When you look up a z-score in a z-table, you're reading the CDF: the total area under the curve to the left of that z-score.

A normal probability plot (or Q-Q plot) helps you assess whether a dataset follows a normal distribution. If the points fall roughly along a straight line, the data is approximately normal. Clear curvature or systematic departures from the line suggest non-normality.

The central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the original population distribution. This is why the normal distribution shows up so frequently in statistics: even when individual data isn't normal, averages of samples tend to be.