14.6 Descriptive Statistics (Graphical)

2 Objectives ► Data in Categories ► Histograms and the Distribution of Data ► The Normal Distribution

3 Data in Categories

4 Numerical data is the data whose values are real numbers. For example, age, height, and cholesterol level are real numbers. But, in many situations we obtain categorical data – that is, the data tell us only whether an individual does or does not belong to a particular category. For example, a survey of your class could include categorical data such as hair color (black, brown, blond, red), political affiliation (Democrat, Republican, Independent), and so on.

5 Data in Categories In this case, the data indicate the number of individuals in each category. Categorical data can be represented by bar graphs or pie charts. A bar graph consists of vertical bars, one bar for each category. The height of each bar is proportional to the number of individuals in that category. So, the y-axis has a numerical scale corresponding to the number (or the proportion) of the individuals in each category. The labels on the x-axis describe the different categories.

6 Data in Categories U.S. energy sources

7 Data in Categories A pie chart consists of a circle divided into sectors, one sector for each category. The central angle of each sector is proportional to the number of individuals in that category. Each sector is labeled with the name of the corresponding category. U.S. energy sources

8 Histogram and the Distribution of Data

9 Histograms and the Distribution of Data In general, to put numerical data into categories, we divide the range of the data into non-overlapping contiguous intervals of equal lengths, called bins (bars). To draw a histogram of the data, we first label the bins on the x-axis and then erect a rectangle (bar) on each bin; the height (and hence also the area) of each rectangle is proportional to the number of data points in that bin.

10 Example 1 – Drawing a Histogram Hospitals routinely measure patients’ heart rates. The list below gives the heart rate (in beats per minute) of 40 patients admitted to the ICU unit at a city hospital. Draw a histogram of the heart rate data. Use bins of size 10 starting at 50 beats per minute.

11 Example 1 – Drawing a Histogram First, we organize the data into bins of size 10 by using a frequency table:

12 Example 1 – Drawing a Histogram Next, we set up our histogram and graph a bar over each interval:

13 Example 1 – Solution Frequency Distribution Histogram of heart rate data

14 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 the figures below. Symmetric and skew distributions

15 Histograms and the Distribution of Data The previous heart rate data 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.

16 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.

17 The Normal Distribution

18 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. The standard normal distribution (or standard normal curve) is modeled by the function:

19 The Normal Distribution This distribution has mean 0 and standard deviation 1. Normal distributions with different means and standard deviations are modeled by transformations (shifting and stretching) of the previous function. Specifically, the normal distribution with mean  and standard deviation  is modeled by the function:

20 The Normal Distribution All normal distributions have the same general shape, called a bell curve. Graphs of several normal distributions are shown. Normal curves

21 The Normal Distribution It can be shown by using calculus that for any data that are normally distributed, about 68% of the data are within one standard deviation of the mean, about 95% are within two standard deviations of the mean, and about 99.7% are within three standard deviations of the mean. In fact, for normally distributed data, the proportion of the data between any two values is completely determined by the mean and standard deviation of the data.

22 The Normal Distribution

23 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

24 Example 3(a) – Solution IQ scores between 85 and 115 are within one standard deviation of the mean: 100 – 15 = 85 and = 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.

25 Example 3(b) – Solution IQ scores between 70 and 130 are within two standard deviations of the mean: 100 – 2(15) = 70 and (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).

26 Example 3(b) – Solution So, the probability that a randomly selected individual has an IQ of at least 130 is

27 Example 3(c) – Solution 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

28 Data in Categories Practice: p #1, 2, 3-19o, 23-31o