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Probability and Statistics Copyright © Cengage Learning. All rights reserved. Descriptive Statistics 14.6 (Graphical) Copyright © Cengage Learning. All rights reserved. Objectives ► Histograms and the Distribution of Data ► The Normal Distribution Histogram and the Distribution of Data Example 1 – Describe what this histogram tells you. Histogram of heart rate data Figure 2 cont’d Histograms and the Distribution of Data A histogram gives a visual representation of how the data are distributed in the different bins. The histogram allows us to determine whether the data are symmetric about the mean, as in Figure 3. Symmetric and skew distributions Figure 3 Histograms and the Distribution of Data The heart rate data in Figure 2 are approximately symmetric. If the histogram has a long “tail” on the right, we say that the data are skewed to the right. Similarly, if there is a long tail on the left, the data are skewed to the left. Since the area of each bar in the histogram is proportional to the number of data points in that category, it follows that the median of the data is located at the x-value that divides the area of the histogram in half. Histograms and the Distribution of Data The extreme values have a large effect on the mean but not on the median. So if the data are skewed to the right, the mean is to the right of the median; if the data are skewed to the left, the mean is to the left of the median. The Normal Distribution Suppose we measured the right foot length of 30 teachers and graphed the results. Assume the first person had a 10 inch foot. We could create a bar graph and plot that person on the graph. Number of People with that Shoe Size If our second subject had a 9 inch foot, we would add her to the graph. As we continued to plot foot lengths, a pattern would begin to emerge. 8 7 6 5 4 3 2 1 . 4 5 6 7 8 9 10 11 12 Length of Right Foot 13 14 Number of People with that Shoe Size Notice how there are more people (n=6) with a 10 inch right foot than any other length. Notice also how as the length becomes larger or smaller, there are fewer and fewer people with that measurement. This is a characteristics of many variables that we measure. There is a tendency to have most measurements in the middle, and fewer as we approach the high and low extremes. 8 7 6 5 4 3 2 1 If we were to connect the top of each bar, we would create a frequency polygon. . 4 5 6 7 8 9 10 11 12 Length of Right Foot 13 14 Number of People with that Shoe Size You will notice that if we smooth the lines, our data almost creates a bell shaped curve. 8 7 6 5 4 3 2 1 4 5 6 7 8 9 10 11 12 13 14 Length of Right Foot You will notice that if we smooth the lines, our data almost creates a bell shaped curve. Number of People with that Shoe Size This bell shaped curve is known as the “Bell Curve” or the “Normal Curve.” 8 7 6 5 4 3 2 1 4 5 6 7 8 9 10 11 12 13 14 Length of Right Foot Number of Students Whenever you see a normal curve, you should imagine the HISTOGRAM within it. 9 8 7 6 5 4 3 2 1 12 13 14 15 16 17 18 19 20 21 22 Points on a Quiz The Normal Distribution Most real-world data are distributed in a special way called a normal distribution. For example, the heart rate data in Example 1 are approximately normally distributed. That is, the shape of the distribution is relatively symmetric, centered at the mean of the distribution with extreme values at either tail. Can you think of other measures that might have a Normal Distribution? Properties of Normal Distributions Normal distribution • The most important continuous probability distribution in statistics. • The graph of a normal distribution is called the normal curve. x 16 The Normal Distribution All normal distributions have the same general shape, called a bell curve. Graphs of several normal distributions are shown in Figure 5. Normal curves Figure 5 Means and Standard Deviations • A normal distribution can have any mean and any positive standard deviation. • The mean gives the location of the line of symmetry. • The standard deviation describes the spread of the data. 18 μ = 3.5 σ = 1.5 μ = 3.5 σ = 0.7 μ = 1.5 σ = 0.7 The 68-95-99.7 Rule • Normal models give us an idea of how extreme a value is by telling us how likely it is to find one that far from the mean. • We can find these numbers precisely, but until then we will use a simple rule that tells us a lot about the Normal model… Slide 6- 19 The 68-95-99.7 Rule (cont.) • It turns out that in a Normal model: – about 68% of the values fall within one standard deviation of the mean; – about 95% of the values fall within two standard deviations of the mean; and, – about 99.7% (almost all!) of the values fall within three standard deviations of the mean. Slide 6- 20 The 68-95-99.7 Rule (cont.) • The following shows what the 68-95-99.7 Rule tells us: Slide 6- 21 The Normal Distribution Example 3 – Using the Normal Distribution (Empirical Rule) IQ scores are normally distributed with mean 100 and standard deviation 15. Find the proportion of the population with IQ scores in the given interval. Also find the probability that a randomly selected individual has an IQ score in the given interval. (a) Between 85 and 115 (b) At least 130 (c) At most 130 23 Example 3(a) – Solution IQ scores between 85 and 115 are within one standard deviation of the mean: 100 – 15 = 85 and 100 + 15 = 115 By the Empirical Rule, about 68% of the population have IQ scores between 85 and 115. So the probability that a randomly selected individual has an IQ between 85 and 115 is 0.68. 24 Example 3(b) – Solution cont’d IQ scores between 70 and 130 are within two standard deviations of the mean: 100 – 2(15) = 70 and 100 + 2(15) = 130 By the Empirical Rule, about 95% of the population have IQ scores between 70 and 130. The remaining 5% of the population have IQ scores above 130 or below 70. Since normally distributed data are symmetric about the mean, it follows that 2.5% have IQ scores above 130 (and 2.5% below 70). 25 Example 3(b) – Solution cont’d So the probability that a randomly selected individual has an IQ of at least 130 is 0.025. 26 Example 3(c) – Solution cont’d By part (b), 2.5% of the population have IQ scores above 130. It follows that the rest of the population have IQ scores below 130. Thus 97.5% of the population have IQ scores of 130 or below. So the probability that a randomly selected individual has an IQ of 130 at most is 0.975. 27