Download 1 The distribution of a variable tells us what values the variable

5.8 Normal Distributions
We now have a number of graphical and numerical tools for describing distributions. We also have a clear strategy for exploring data on a single numerical variable.
1. Plot the data: make a graph, usually a histogram, stemplot, or boxplot.
2. Look for the overall pattern (shape, center, variability) and for striking deviations such as outliers.
3. Calculate a numerical summary to give some description of center and variability.
Today we add a 4th step:
4. If the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve, then draw that curve superimposed on the histogram. The following histogram shows the Iowa Test vocabulary scores of 947 7th­grade students. Like most histograms from national standardized tests, the histogram is symmetric, single­peaked, and has a distinctive bell shape. The curve superimposed on the histogram gives a picture of the overall pattern but ignores minor irregularities as well as any outliers. The curve shown below is a normal curve. A distribution whose shape is described by a normal curve is a normal distribution.
The distribution of a variable tells us what values the variable takes and how often it takes these values. A normal distribution is described by a normal curve. The area under the curve above any interval of values tells us what proportion of all values of the variable lie in that interval. The total area under the curve is exactly 1. If you think of the bars as representing 100% of the population, then 100% of the population is under the curve, and 100%, represented as a decimal, is 1.
The normal curve is a good approximation of the real­life distribution for a variety of biological measures, such as height, weight, heart rate, blood pressure, etc., for a particular species and gender. The figure below (left) shows the heights of American women. The peak of the curve gives us the center of the data, which is the mean as well as the median, since half of the values fall on each side of the center in a normal curve. The figure on the right shows heights of women and men. Note that both fit the normal distribution. 1
The specific form of a normal distribution and the role it plays in statistical theory are very special. All normal curves have the same general shape. They are symmetric and bell­shaped, with tails that fall down rapidly from a central peak. The center of the normal curve is the mean of the distribution as well as the median. Normal curves have the special property that their variability is determined completely by a single number, the standard deviation. The graph below (left) shows two normal curves with the same mean but different standard deviations. The graph below (right) shows two normal curves with the same standard deviation but different means.
The mean of a normal distribution is at the center of symmetry of the normal curve. The standard deviation of a normal distribution is the distance from the center to the change­of­curvature points on either side. The quartiles of a normal distribution are located about 0.67 (which is about 2/3) of a standard deviation away from the mean. In particular, the first quartile (Q1)is located at 0.67 standard deviation below the mean and, by symmetry, the third quartile (Q3) is located at 0.67 standard deviation above the mean. This is illustrated in the graph below.
Example 1: Suppose the mean (μ) of a normal distribution is 70, and the standard deviation (σ) is 10. a) Determine the values of Q1 and Q3.
b) Determine the range of the middle 50% of the data.
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