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In statistics, a normal distribution is a type of continuous probability distribution for a real-valued random variable. It is symmetrical, meaning it forms a perfectly bell-shaped curve. This distribution is centered around the mean, which is the average of all data points, and demonstrates how every point in the dataset clusters around the mean. The spread of data points is characterized by the standard deviation.
The normal distribution is important because it describes many natural phenomena. Variables like test scores, heights, and in this case, blood pressure readings, tend to follow a normal distribution. This concept allows us to make predictions about where a certain percentage of information will lie once the data is plotted.

• Symmetrical bell-shaped curve.
• Mean, median, and mode are all equal.
• The total area under the curve equals 1.

A z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It tells us how many standard deviations away a particular score is from the mean. The formula to find the z-score is:
\[z = \frac{X - \mu}{\sigma}\]
* Where \(X\) is the individual score,* \(\mu\) is the mean of the scores,* \(\sigma\) is the standard deviation.
Using z-scores, we can standardize our data, making it possible to compare scores from different datasets. In the given exercise, the z-score calculation for a blood pressure reading of 140 helps determine how unusual this reading is compared to the average of 121.

• Z-score of 0 means the value is exactly the mean.
• Positive z-scores indicate a value above the mean.
• Negative z-scores indicate a value below the mean.

Percentiles are measures that divide a dataset into 100 equal parts. They indicate the relative standing of a data point in this dataset. For example, if your blood pressure reading is at the 90th percentile, it means you score higher than 90% of the other readings.
Percentiles take advantage of the cumulative areas under the normal distribution curve. To determine percentiles, such as the 90th percentile in the exercise, you look up the z-score that corresponds to that percentile and use this to find your desired value.

• Percentiles help compare your score to a larger group.
• Provide information about the spread and concentration of the data.
• Routinely used in scoring standardized tests or physical measures.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It is crucial in the context of a normal distribution because it affects the shape of the distribution curve. It tells us how spread out the numbers are in a dataset.
A low standard deviation means that most of the numbers are close to the mean, while a high standard deviation means that the numbers are more spread out.
In the exercise, a standard deviation of 16 indicates the average distance of blood pressure readings from the mean reading of 121.

• Essential in calculating z-scores.
• Helps determine the variability in data.
• A standard deviation of 0 indicates no variation.