Chapter 3 Numerical Descriptions of Data 75 Chapter 3.docx
- 1. Chapter 3: Numerical Descriptions of Data 75 Chapter 3: Numerical Descriptions of Data Chapter 1 discussed what a population, sample, parameter, and statistic are, and how to take different types of samples. Chapter 2 discussed ways to graphically display data. There was also a discussion of important characteristics: center, variations, distribution, outliers, and changing characteristics of the data over time. Distributions and outliers can be answered using graphical means. Finding the center and variation can be done using numerical methods that will be discussed in this chapter. Both graphical and numerical methods are part of a branch of statistics known as descriptive statistics. Later descriptive statistics will be used to make decisions and/or estimate population parameters using methods that are part of the branch called inferential statistics. Section 3.1: Measures of Center This section focuses on measures of central tendency. Many times you are asking what to expect on average. Such as when you pick a major, you would probably ask how much you expect to earn in that field. If you are thinking of relocating to a new town,
- 2. you might ask how much you can expect to pay for housing. If you are planting vegetables in the spring, you might want to know how long it will be until you can harvest. These questions, and many more, can be answered by knowing the center of the data set. There are three measures of the “center” of the data. They are the mode, median, and mean. Any of the values can be referred to as the “average.” The mode is the data value that occurs the most frequently in the data. To find it, you count how often each data value occurs, and then determine which data value occurs most often. The median is the data value in the middle of a sorted list of data. To find it, you put the data in order, and then determine which data value is in the middle of the data set. The mean is the arithmetic average of the numbers. This is the center that most people call the average, though all three – mean, median, and mode – really are averages. There are no symbols for the mode and the median, but the mean is used a great deal, and statisticians gave it a symbol. There are actually two symbols, one for the population parameter and one for the sample statistic. In most cases you cannot find the population parameter, so you use the sample statistic to estimate the population parameter.
- 3. Population Mean: µ = Σx N , pronounced mu N is the size of the population. x represents a data value. x∑ means to add up all of the data values. Chapter 3: Numerical Descriptions of Data 76 Sample Mean: x = Σx n , pronounced x bar. n is the size of the sample. x represents a data value. x∑ means to add up all of the data values. The value for x is used to estimate µ since µ can’t be calculated in most situations.
- 4. Example #3.1.1: Finding the Mean, Median, and Mode Suppose a vet wants to find the average weight of cats. The weights (in pounds) of five cats are in table #3.1.1. Table #3.1.1: Weights of cats in pounds 6.8 8.2 7.5 9.4 8.2 Find the mean, median, and mode of the weight of a cat. Solution : Before starting any mathematics problem, it is always a good idea to define the unknown in the problem. In this case, you want to define the variable. The symbol for the variable is x. The variable is x = weight of a cat
- 5. Mean: x = 6.8+8.2+ 7.5+ 9.4+8.2 5 = 40.1 5 = 8.02 pounds Median: You need to sort the list for both the median and mode. The sorted list is in table #3.1.2. Table #3.1.2: Sorted List of Cats’ Weights 6.8 7.5 8.2 8.2 9.4 There are 5 data points so the middle of the list would be the
- 6. 3rd number. (Just put a finger at each end of the list and move them toward the center one number at a time. Where your fingers meet is the median.) Table #3.1.3: Sorted List of Cats’ Weights with Median Marked 6.8 7.5 8.2 8.2 9.4 Chapter 3: Numerical Descriptions of Data 77 The median is therefore 8.2 pounds. Mode: This is easiest to do from the sorted list that is in table #3.1.2. Which value appears the most number of times? The number 8.2 appears twice,
- 7. while all other numbers appear once. Mode = 8.2 pounds. A data set can have more than one mode. If there is a tie between two values for the most number of times then both values are the mode and the data is called bimodal (two modes). If every data point occurs the same number of times, there is no mode. If there are more than two numbers that appear the most times, then usually there is no mode. In example #3.1.1, there were an odd number of data points. In that case, the median was just the middle number. What happens if there is an even number of data points? What would you do? Example #3.1.2: Finding the Median with an Even Number of Data Points Suppose a vet wants to find the median weight of cats. The
- 8. weights (in pounds) of six cats are in table #3.1.4. Find the median Table #3.1.4: Weights of Six Cats 6.8 8.2 7.5 9.4 8.2 6.3