NUMERICAL DESCRIPTIVE MEASURES CHAPTER 4 NUMERICAL DESCRIPTIVE MEASURES

4.1 MEASURES OF CENTRAL TENDENCY In Chapter 3, we used tables and graphs to summarize a data set. In Chapter 4, we will estimate numerical summary measures to identify important features of a distribution. We begin by focusing on numerical summary measures that identify the center and spread of a distribution. Measure of Central Tendency Measure of central tendency tells us where the center of a histogram or a frequency distribution lies. We will focus on three measures of central tendency: Mean Median Mode Other measures include trimmed mean, weighted mean, & geometric mean

Mean or Arithmetic Mean The mean or arithmetic for ungrouped or raw data is defined as the sum of all values divided by the number of values in the data set. So, Mean for population data: Mean for sample data:

Mean or Arithmetic Mean Example 1 – Sample Mean The following table gives the standard deductions and personal exemptions for persons filing with “single” status on their 2009 state income taxes in a random sample of 9 states. Find the mean for the data on standard deduction.

Mean or Arithmetic Mean Example 2 – Population Mean The following data set belongs to a population 5 -7 2 0 -9 16 10 7 Find the mean.

Mean or Arithmetic Mean Example 3 – Effect of outliers on Mean Find the mean for the data on Example 1 for personal exemption without the states of Minnesota, North Dakota, Rhode Island, and Vermont. Now, find the mean for the data on Problem 3.11 for personal exemption. 6

Mean or Arithmetic Mean - Summary Each value of the data set is used in the calculation. The population mean µ is constant, whereas the sample mean varies from sample to sample. Mean is not always the best measure of central tendency of a data set. Mean is greatly affected by outliers. When outliers exist in a data set, it is important to use trimmed mean or median. Trimmed mean is calculated by dropping a certain percentage of values from both ends of a ranked data set. 7

Median The median is the value of the middle term in a data set that has been arranged in increasing order. The steps for calculating median are: Arrange the data set in increasing order. Find or locate the middle term. Then the value of this term is the median. To locate the middle term and find median: For odd number of observations, location of middle term is Thus, median = Value of middle term For even number of observations, location of the middle term is based on two terms, one from the left and other from the right of the data set. Thus, median =

Median Example 4 Find the median for the data on Example 1 for standard deduction. First, we rank the given data in increasing order as follows: 1865 2000 2100 3000 3250 5450 5450 5450 5450

Median Example 5 Find the median for: 258.7 77.8 393.1 427.0 273.6 2977.0 First, we rank the given data in increasing order as follows: 77.8 258.7 273.6 393.1 427.0 2977.0 10

Median - Summary Median gives the middle of a distribution, with half the data values to the left of the median and half to the right of the median. Median is not influenced by outliers. Median is preferred over the mean as a measure of central tendency for data sets that contain outliers. 11

Mode Mode is defined as the value that occurs the most or with the highest frequency in a data set. Example 6 Find the mode for the data on Example 1 for standard deduction. In this data set, 5450 occurs four times while each remaining values occurs only once. 5450 is the mode because it has the highest frequency. Therefore, Mode = $5450

Mode - Summary Mode can be calculated for both qualitative and quantitative data set. A data set may have no or more than one mode. No mode = Data set where each value occurs only once. One mode = Data set where there is only one value with the highest frequency. This data set is called unimodal. Two modes = Data set where there are two values with the highest frequencies. This data set is called bimodal. More than two modes = Data set where there are more than two values with the highest frequencies. This data set is called multimodal. 13

Relationships among the Mean, Median, and Mode For a symmetric histogram and frequency distribution curve mean = median = mode For right-skewed histogram and frequency distribution curve mode < median < mean For left-skewed histogram and frequency distribution curve mean < median < mode

MEASURES OF DISPERSION FOR UNGROUPED DATA Mean, median, or mode does not tell us the spread, variation, or dispersion of a distribution. For example: The number of car thefts that occurred in two neighboring cities for the past 12 days are given as: City A: 6 4 7 11 4 3 9 7 2 7 9 15 City B: 8 10 14 0 0 10 20 0 15 3 3 1 The data sets have the same mean, 7 cars per day. Without the data set, this suggests that the same number of cars were stolen per day for the past 12 days in both cities. Using a Dotplot, the two cities have different variation.

4.2 Measures of Spread Range Variance Standard deviation We need a measure of dispersion or variation Range Variance Standard deviation = Largest Value – Smallest Value

Range for Ungrouped Data Example 7 The following data give the number of pieces of junk mail received by 7 families during the past month. 41 33 28 21 29 19 2 a. Find the range with all the values in the data set b. Find the range without the value of 2

Range - Summary Range is not a good measure of dispersion of a data set with outliers because its value is greatly affected by outliers. Range is also not a satisfactory measure of dispersion because it uses only two values, largest and smallest, in the data set. 18

Variance and Standard Deviation The standard deviation is the most used measure of dispersion because it tells the closeness of the values of a data set to or around the mean. Variance is denoted as (σ sigma) σ2 for population data s2 for sample data Standard deviation is defined as the principal square root of the variance Standard deviation is denoted as σ for population data s for sample data What does a value of the standard deviation mean? Lower value = Values are spread relatively over a smaller range around the mean Larger value = Values are spread relatively over a larger range around the mean

Variance and Standard Deviation – Formula for Ungrouped Data Basic Formula Short-Cut Formula Variance Standard Deviation Note indicates the deviation of each value of the data set from the mean. The sum of all the deviations must always be zero.

Variance and Standard Deviation Example 8 - Sample Find the variance and standard deviation for the sample data in the given table.

Variance and Standard Deviation - Summary The values of the variance and the standard deviation cannot be negative. Why? The value of variance and standard deviation can be zero, if a data set has no variation. The measurement unit of variance is the square of the measurement unit of the original data. The measurement unit of standard deviation is the measurement unit of the original data. Why? 22

Population Parameters and Sample Statistics Mean, median, mode, range, variance, or standard deviation calculated for: A population data set is called a population parameter or just parameter. µ and σ are examples of population parameters A sample data set is called a sample statistic, or just statistic. are example of sample statistic.

USE OF STANDARD DEVIATION So far, we can find the mean and standard deviation of a distribution data. But the question is: Whether we can use the mean and standard deviation to find the percentage or proportion of the data set that lie within an interval of the mean. The answer is yes if we combine the mean and standard deviation. To do this, we can use Chebyshev’s theorem or Empirical rule. Our focus is only on the empirical rule

Empirical Rule Empirical rule only works for a bell-shaped distribution. That is, empirical rule cannot be applied to other distributions such as left-skewed, right-skewed, and uniform distributions. For a bell-shaped distribution, the percentage or proportion of a data set that lie within an interval of the mean is determined under the following three rules 68% of the observations lie within one standard deviation of the mean 95% of the observations lie within two standard deviations of the mean 99.7% of the observations lie within three standard deviations of the mean.

Empirical Rule Example 12a Solution Suppose that on a certain section of I-95 with a posted speed limit of 65 mph, the speeds of all vehicles have a bell-shape distribution with a mean of 72 mph and a standard deviation of 3 mph. Using the empirical rule, find the percentage of vehicles with 63 to 81 mph on this section of I-95. Solution x

Empirical Rule Example: The prices of all college textbooks follow a bell-shaped distribution with a mean of $105 and a standard deviation of $20. A) Find the percentage of all college textbooks with thier prices between $85 and $125 Solution:

Empirical Rule B) Find the percentage of all college textbooks with thier prices between $65 and $145 C) Find the interval that contains the prices of 99.7%.

4.3 Boxplots Definition A measure of position determines the position of a single value in relation to other values in a sample or population. We will discuss only the following measures of position. Quartiles and Interquartile Range Percentiles and Percentile Rank

Quartiles and Interquartile Range Definition Quartiles are three summary measures that divide a ranked data set into four equal parts. The first quartile is the value of the middle term among the observations that are less than the median The second quartile is the same as the median of a data set. The third quartile is the value of the middle term among the observations that are greater than the median.

Quartiles and Interquartile Range Calculating Interquartile Range Interquartile range is the difference between the third and first quartiles. That is, IQR = Interquartile range = Q3 – Q1

Example 13 The 2008 profits (rounded to billions of dollars) of 12 companies selected from all over the world are shown in the table. Find the values of the three quartiles. Where does the 2008 profits of Merck & Co fall in relation to these quartiles? Find the interquartile range.

Example 13 a) solution b) solution

Box-and-whisker plot use the Median 1st quartile, 3rd quartile, and Boxplots Box-and-whisker plot use the Median 1st quartile, 3rd quartile, and Smallest and largest values in the data set between the lower and upper inner fences to graphically display data. Lower inner fence = 1.5(IQR) below the Q1 = Q1 - 1.5(IQR) Upper inner fence = 1.5(IQR) above the Q3 = Q3 + 1.5(IQR) Advantages of box-and-whisker plot Visually displays the center, spread, and the skewness of a data set. Clearly identifies outliers. Helps to compare different distributions.

Box-and-Whisker Plot Steps to Plot Box-and-Whisker Chart Arrange the data set in increasing order Calculate the following: Median, Q1, Q3, and IQR = Q3 - Q1 Determine the lower and upper inner fences Determine the smallest and largest values within the lower and upper inner fences. Draw a horizontal number line and mark the line covering all the values in the data set. Above the number line, draw a box with The left side at Q1 and the right side at Q3 and A vertical line at the median (inside the box). Identify the smallest and largest values within the lower and upper inner fences with short vertical lines above the number line. Then , draw two lines joining each vertical line to the box. These lines are called whiskers.

Box-and-Whisker Plot Steps to Plot Box-and-Whisker Chart A value that falls outside either of the inner fences is called an outlier. An outlier could be: Mild or Extreme A mild outlier occurs when a value falls outside any of the inner fences but inside either a lower or upper outer fence. An extreme outlier is a value that falls outside either of the outer fences. Calculating outer fences: Lower outer fence = 3(IQR) below Q1 = Q1 - 3(IQR) Upper outer fence = 3(IQR) above Q3 = Q3 + 3(IQR)

Example 16 The following data are the incomes (in thousands of dollars) for a sample of 12 households. 75 69 84 112 74 104 81 90 94 144 79 98 Construct a box-and-whisker plot for these data. Step 1 & 2

Example 16 Step 3. Step 4.

Example 16 Step 5-8.

Example 16 Is this a Mild or extreme? Calculating outer fences: Lower outer fence = 3(IQR) below Q1 = Q1 - 3(IQR) Upper outer fence = 3(IQR) above Q3 = Q3 + 3(IQR) =101+3(24)=173 which is a Mild Outlier