6.2 Using the Normal Distribution

Normal Distribution Calculations and Interpretation

The normal distribution gives you a way to calculate the probability of specific outcomes, like the chance a randomly selected test score falls above or below a certain value. To do this, you convert raw values into z-scores, then use tables or technology to find the corresponding probabilities.

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Probability Calculations in Normal Distributions

The total area under any normal curve equals 1 (or 100%), representing all possible outcomes. To find the probability associated with a specific value, you standardize it using the z-score formula:

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

• $x$ = the value you're interested in (e.g., a specific test score)
• $\mu$ = the mean of the distribution
• $\sigma$ = the standard deviation of the distribution

The z-score tells you how many standard deviations $x$ is from the mean. A z-score of 1.5 means the value sits 1.5 standard deviations above the mean; a z-score of -2 means it's 2 standard deviations below.

Once you have the z-score, here's how to find probabilities:

1. Look up the z-score in a standard normal distribution table (or use technology). The table gives you the area to the left of that z-score.
2. That area is the probability that a randomly selected value is less than or equal to $x$.
3. To find the area to the right, subtract the left-side area from 1. For example, if the area to the left is 0.8413, the area to the right is $1 - 0.8413 = 0.1587$.
4. Express the result as a decimal (0.1587) or a percentage (15.87%).

This left-side lookup is called the cumulative distribution function (CDF): it gives the probability that a random variable is less than or equal to a given value.

Interpretation of Shaded Areas

When you see a shaded region on a normal curve, that shaded area is the probability of a value falling in that region. There are three common setups:

• Area to the left of a z-score: the probability of a value being less than or equal to the corresponding $x$-value. For example, the probability of scoring below 75 on an exam.
• Area to the right of a z-score: the probability of a value being greater than the corresponding $x$-value. For example, the probability of scoring above 90.
• Area between two z-scores: the probability of a value falling within a specific range. You calculate this by finding the left-side area for each z-score and subtracting the smaller from the larger.

The empirical rule (68-95-99.7 rule) provides quick estimates without any table lookups:

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

This is useful for ballpark checks, but for precise probabilities you'll still need the z-table or technology.

Using Technology for Normal Distribution Probabilities

Statistical software and calculators speed up these calculations and reduce errors. Common tools include graphing calculators (like the TI-84), spreadsheet software (like Excel or Google Sheets), and online z-score calculators.

General steps for using technology:

1. Identify the parameters you need: the mean ($\mu$), standard deviation ($\sigma$), and the value or range of interest.
2. Enter these into the appropriate function. On a TI-84, that's normalcdf for area between two values or normalpdf for the density at a point. In Excel, use =NORM.DIST(x, mean, sd, TRUE) for the cumulative probability.
3. Read the output as the probability associated with your value or range.
4. Cross-check against a manual z-table calculation when possible, especially while you're still learning.

The inverse normal function works in the other direction: you input a probability, and it returns the corresponding $x$-value. This is useful for questions like "What score do you need to be in the top 10%?" On a TI-84, use invNorm; in Excel, use =NORM.INV(probability, mean, sd).

Advanced Concepts in Normal Distributions

• The central limit theorem (CLT) states that the distribution of sample means approaches a normal distribution as the sample size increases, even if the original population isn't normal. This is why the normal distribution shows up so often in statistics.
• Skewness describes asymmetry in a distribution. A perfectly normal distribution has zero skew. Positive skew means a longer right tail; negative skew means a longer left tail. When data is noticeably skewed, the normal model may not fit well.
• Kurtosis describes how heavy the tails are compared to a normal distribution. High kurtosis means more extreme outliers; low kurtosis means lighter tails.
• Standardizing values with z-scores lets you compare data from different normal distributions directly. A z-score of 1.2 on one exam means the same relative position as a z-score of 1.2 on a completely different exam, regardless of the original scales.