Math in Society: Mathematics for liberal arts majors

There is a pattern or shape that comes up frequently when graphing data called the Normal distribution. You might be familiar with it as the bell curve. When we take measurements like the lengths of newborn babies or the weights of potato chip bags on an assembly line there are usually more measurements near the mean and fewer as we get further away from the mean on each side. We often see histograms with this shape:

Notice how the shape is unimodal and symmetric with larger and smaller values getting less and less common as they get further from the mean. When we take samples from a population we might not get an exact Normal distribution but it is often close. We call this approximately Normal or nearly Normally distributed. The curve drawn over the histogram shows that the data are nearly Normal. Earlier in this chapter we learned about the mean, median and mode as measures of center and the standard deviation as a measure of variation or spread. The mean is often written with the greek letter mu, \(\mu\text{.}\) The standard deviation is usualy represented by the lowercase Greek letter sigma, \(\sigma\text{.}\)

Now let’s look at an example where we draw and label a Normal distribution given the mean and standard deviation

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All of the Normal drawings in this section were created with the online probability calculator at onlinestatbook.com/2/calculators/normal_dist.html which is in the public domain.

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No matter what the mean and standard deviation are for a Normal Distribution, they always have the same shape or distribution of the data. We can describe this with the Empirical Rule which is also called the 68-95-99.7 Rule.

The numbers in the 68-95-99.7 rule describe the percentage of data or area within 1, 2 and 3 standard deviations of the mean. Let’s look at our previous example with scores on a math quiz that are approximately normally distributed with a mean of 18 points and a standard deviation of 4 points.

Now you can try this next example:

Now that we have learned about the Empirical rule we can use it to find approximate probabilities. First we will calculate the percentage in each segment of the Normal distribution. Returning to our example of quiz scores with a mean of 18 points and a standard deviation of 4 points, we can divide the curve into segments by drawing a line at each standard deviation.

We can summarize all the calculations in this drawing:

Now let’s continue our quiz score example to see how to use the Empirical Rule to find probabilities.

Now you can try one with this next example.

The Empirical rule is used to find approximate probabilities and it only works if the number is one of the labels on our scale. We usually want to calculate an exact answer with any value. We will use technology to calculate probabilities with two different methods: Onlinestatbook.com and spreadsheets. Check with your class notes or instructor on which technology to use.

To use a spreadsheet, we will use the spreadsheet formula

Now you can this try this example.

Earlier in the chapter we learned how to calculate a Z-score, which is a way to determine how usual or unusual a data value is, or compare values from two different distributions. To calculate the Z-score for a data value, we calculate its distance from the mean. Then we divide that distance by the standard deviation to see how many standard deviations that is. Thus, the Z-score of a data value is the number of standard deviations it is away from the mean.

If a data value is equal to the mean it will have a Z-score of 0. If a data value is one standard deviation above the mean, it will have a Z-score of 1. We can make a Normal distribution of Z-scores and it will have a mean of 0 and a standard deviation of 1. This is called the Standard Normal distribution, shown below.

Applying the Empirical Rule in percentage form to the Standard Normal curve looks like this.

Sometimes we may want to use the decimal form of the numbers instead.

Applying the Empirical Rule to the Standard Normal distribution, we know that 68% of all Z-scores will be between -1 and 1, 95% of all Z-scores will be between -2 and 2 and 99.7% of all Z-scores will be between -3 and 3. A Z-score below -3 or above 3 is possible, but is very unlikely.

Now that we have learned more about the Normal distribution, we can come back to confidence intervals which we studied in section 3.1. When we take a sample, our goal is to infer our statistic to the larger population. Let’s say we wanted to know the average height of all 10-year-old kids living in the United States. It would take too much time and money to measure them all so we take a random sample that is representative of the population of 10-year-olds.

If we measured 25 randomly selected 10-year-olds and calculated the sample mean of 54.5 inches, that statistic is called a point estimate. It is a single value that we got from a sample. However, if we measured another random sample of 25 different kids we would likely get a different mean. That is called sampling variation or sampling error. To be more accurate in our estimation of the height of all 10-year olds we give a range of values called a confidence interval.

To construct a confidence interval we need to know the point estimate, the standard error of the data and the desired confidence level. The standard error is an estimate for the standard deviation that we calculate from the sample data and the sample size. You can learn more details about sampling distributions, why they are Normally distributed and how to calculate the standard error in a statistics course.

The most common confidence level is the 95% confidence level. The 95% confidence interval represents the middle 95% of the Normal curve. From the Empirical rule we saw that approximately 95% of the data values are within 2 standard deviations of the mean. Using more exact methods, the middle 95% is 1.96 standard deviations from the mean.

Let’s say that our point estimate for the height of 10-year-olds is 54.5 inches and the standard error is 1.75 inches. To estimate the middle 95% of values we multiply the standard error by 1.96 standard deviations.

The 95% confidence interval is 51.07 to 57.93 inches or (51.07, 57.93) inches.

To interpret a confidence interval we say: We are 95% confident that the true average height of all 10-year-olds is between 51.07 and 57.93 inches. This means if repeated samples were taken and the 95% confidence interval was computed for each sample, about 95% of the intervals would contain the true population mean. About five percent of the intervals would not contain the population mean.

Since the point estimate is our best guess for the value of the parameter, it makes sense to build the confidence interval around that value. The standard error, which is the measure of the uncertainty associated with the point estimate, provides a guide for how large we should make the confidence interval. When the sampling distribution of a point estimate can reasonably be modeled as Normal, the point estimate we observe will be within 1.96 standard errors of the true value of interest. We can be 95% confident this interval captures the true value.

The idea behind confidence intervals and margin of error is that any survey, study or poll will always be different from the population because samples vary. The confidence interval and margin of error reflect the sample size and the variation in the sample.