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In statistics, the mean is a measure of central tendency, giving you an idea of the average value in a dataset. For example, in our exercise about cell phone plans in the United States, the mean monthly charge is denoted by \(\mu = \$ 62\). This value represents the average monthly cost of cell phone plans. To draw the normal curve, you should place this mean value at the center of the horizontal axis (x-axis). The bell-shaped curve will be symmetrical around this point. Understanding the mean is crucial because it is the reference point for interpreting the data under a normal distribution.

Standard deviation is a measure of how spread out the values in a dataset are around the mean. In our example, the standard deviation is given by \(\sigma = \$ 18\). This value shows the extent to which cell phone plan charges deviate from the average charge of \(\$ 62\). The greater the standard deviation, the more spread out the values are. Conversely, a smaller standard deviation indicates that the values are closely clustered around the mean.
When you draw a normal distribution curve, you label points on the x-axis by the mean (\(\mu\)) plus and minus the standard deviations (\(\sigma\)):

• First standard deviation: \(\mu \pm \sigma\), in this case, \(\$ 62 \pm \$ 18 = [\$ 44, \$ 80]\)
• Second standard deviation: \(\mu \pm 2\sigma\), here, \(\$ 62 \pm \$ 36 = [\$ 26, \$ 98]\)
• Third standard deviation: \(\mu \pm 3\sigma\), for this example, \(\$ 62 \pm \$ 54 = [\$ 8, \$ 116]\)

These markers help you understand the spread and variability of the data around the mean.

The area under the curve in a normal distribution is essential to understanding the proportion of data points within certain ranges. Each segment of the bell curve corresponds to a probability. For example, the problem states that the area to the left of \(\$ 44\) (less than one standard deviation below the mean) is \(0.1587\), or 15.87%. This area represents the proportion of cell phone plans with charges less than \(\$ 44\) a month.
Here's how to interpret this result:

• Approximately 15.87% of cell phone plans are cheaper than \(\$ 44\) per month.
• If you randomly select a cell phone plan, there's a 15.87% chance that its cost will be below \(\$ 44\) per month.

To find such areas, you typically use standard normal distribution tables or calculators. These tools give you the area (or probability) related to specific z-scores. A z-score tells you how many standard deviations an element is from the mean. In our case, \(\$ 44\) is equivalent to a z-score of \(\frac{\text{\(44-\)62}}{\text{\(18\sigma\)}} = -1\).