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The standard deviation is a measure that tells us how much the numbers in a data set typically differ from the average (mean) of the set. In the context of a normal distribution, it plays a key role in understanding the spread of the distribution.
In our IQ test example, the standard deviation is 16. This means that most IQ scores tend to hover within 16 points of the average IQ score of 100. Having a smaller standard deviation would mean that the IQ scores are closer to the mean, reflecting lesser individual variation, whereas a larger value would mean greater variety in IQ scores.

• It's used to calculate ranges of values where different portions of the data are expected.
• It helps determine how uncommon a particular score is relative to others.

In normal distributions, key intervals are measured using standard deviations from the mean, helping us to quickly see where most of the data lies and make predictions.

A z-score tells us how many standard deviations away a value is from the mean. It converts a particular data point into a standardized form, providing a clear sense of its relative position in the distribution.
To calculate a z-score, subtract the mean from the point and divide the result by the standard deviation. For example, an IQ score of 116 translates to a z-score:\[z = \frac{116 - 100}{16} = 1\]This calculation shows that 116 is one standard deviation above the mean.

• Z-scores help determine probabilities for values in a normal distribution.
• They allow comparison of data points across different distributions by standardizing them.

Using z-scores, we can look up corresponding probabilities in a standard normal distribution table, or use statistical software to find the proportion of data above, below, or between certain values.

The 68-95-99.7 rule, also known as the empirical rule, helps us understand the distribution of data when plotted on a normal curve. It gives us a quick overview of where the majority of data lies relative to the mean.
According to the rule:

• About 68% of values fall within one standard deviation from the mean.
• Roughly 95% are within two standard deviations.
• And nearly 99.7% are within three standard deviations.

Applying this to our IQ distribution, we see that: - 68% of IQ scores fall between 84 (100 - 16) and 116 (100 + 16). - 95% range from 68 (100 - 2×16) to 132 (100 + 2×16). - 99.7% lie between 52 (100 - 3×16) and 148 (100 + 3×16). This rule is handy for quickly estimating which values are common or rare in any normal distribution.