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CHAPTER 3 Data Description © Copyright McGraw-Hill 2004 3-1 Objectives Summarize data using measures of central tendency, such as the mean, median, mode, and midrange. Describe data using the measures of variation, such as the range, variance, and standard deviation. Identify the position of a data value in a data set using various measures of position, such as percentiles, deciles, and quartiles. © Copyright McGraw-Hill 2004 3-2 Objectives (cont’d.) Use the techniques of exploratory data analysis, including boxplots and five-number summaries to discover various aspects of data. © Copyright McGraw-Hill 2004 3-3 Introduction Statistical methods can be used to summarize data. Measures of average are also called measures of central tendency and include the mean, median, mode, and midrange. Measures that determine the spread of data values are called measures of variation or measures of dispersion and include the range, variance, and standard deviation. © Copyright McGraw-Hill 2004 3-4 Introduction (cont’d.) Measures of position tell where a specific data value falls within the data set or its relative position in comparison with other data values. The most common measures of position are percentiles, deciles, and quartiles. © Copyright McGraw-Hill 2004 3-5 Introduction (cont’d.) The measures of central tendency, variation, and position are part of what is called traditional statistics. This type of data is typically used to confirm conjectures about the data. © Copyright McGraw-Hill 2004 3-6 Introduction (cont’d.) Another type of statistics is called exploratory data analysis. These techniques include the the box plot and the five-number summary. They can be used to explore data to see what they show. © Copyright McGraw-Hill 2004 3-7 Basic Vocabulary A statistic is a characteristic or measure obtained by using the data values from a sample. A parameter is a characteristic or measure obtained by using all the data values for a specific population. When the data in a data set is ordered it is called a data array. © Copyright McGraw-Hill 2004 3-8 General Rounding Rule In statistics the basic rounding rule is that when computations are done in the calculation, rounding should not be done until the final answer is calculated. © Copyright McGraw-Hill 2004 3-9 The Arithmetic Average The mean is the sum of the values divided by the total number of values. Rounding rule: the mean should be rounded to one more decimal place than occurs in the raw data. The type of mean that considers an additional factor is called the weighted mean. © Copyright McGraw-Hill 2004 3-10 The Arithmetic Average The Greek letter (mu) is used to represent the population mean. The symbol mean. Assume that data are obtained from a sample unless otherwise specified. x (“x-bar”) represents the sample © Copyright McGraw-Hill 2004 3-11 Median and Mode The median is the halfway point in a data set. The symbol for the median is MD. The median is found by arranging the data in order and selecting the middle point. The value that occurs most often in a data set is called the mode. The mode for grouped data, or the class with the highest frequency, is the modal class. © Copyright McGraw-Hill 2004 3-12 Midrange The midrange is defined as the sum of the lowest and highest values in the data set divided by 2. The symbol for midrange is MR. © Copyright McGraw-Hill 2004 3-13 Central Tendency: The Mean One computes the mean by using all the values of the data. The mean varies less than the median or mode when samples are taken from the same population and all three measures are computed for these samples. The mean is used in computing other statistics, such as variance. © Copyright McGraw-Hill 2004 3-14 Central Tendency: The Mean (cont’d.) The mean for the data set is unique, and not necessarily one of the data values. The mean cannot be computed for an open- ended frequency distribution. The mean is affected by extremely high or low values and may not be the appropriate average to use in these situations. © Copyright McGraw-Hill 2004 3-15 Central Tendency: The Median The median is used when one must find the center or middle value of a data set. The median is used when one must determine whether the data values fall into the upper half or lower half of the distribution. The median is used to find the average of an openended distribution. The median is affected less than the mean by extremely high or extremely low values. © Copyright McGraw-Hill 2004 3-16 Central Tendency: The Mode The mode is used when the most typical case is desired. The mode is the easiest average to compute. The mode can be used when the data are nominal, such as religious preference, gender, or political affiliation. The mode is not always unique. A data set can have more than one mode, or the mode may not exist for a data set. © Copyright McGraw-Hill 2004 3-17 Central Tendency: The Midrange The midrange is easy to compute. The midrange gives the midpoint. The midrange is affected by extremely high or low values in a data set. © Copyright McGraw-Hill 2004 3-18 Distribution Shapes In a positively skewed or right skewed distribution, the majority of the data values fall to the left of the mean and cluster at the lower end of the distribution. © Copyright McGraw-Hill 2004 3-19 Distribution Shapes (cont’d.) In a symmetrical distribution, the data values are evenly distributed on both sides of the mean. © Copyright McGraw-Hill 2004 3-20 Distribution Shapes (cont’d.) When the majority of the data values fall to the right of the mean and cluster at the upper end of the distribution, with the tail to the left, the distribution is said to be negatively skewed or left skewed. © Copyright McGraw-Hill 2004 3-21 The Range The range is the highest value minus the lowest value in a data set. The symbol R is used for the range. © Copyright McGraw-Hill 2004 3-22 Variance and Standard Deviation The variance is the average of the squares of the distance each value is from the mean. The symbol for the population variance is 2. x 2 2 N © Copyright McGraw-Hill 2004 3-23 Variance and Standard Deviation The standard deviation is the square root of the variance. The symbol for the population standard deviation is . Rounding rule: The final answer should be rounded to one more decimal place than the original data. x 2 2 N © Copyright McGraw-Hill 2004 3-24 Coefficient of Variation The coefficient of variation is the standard deviation divided by the mean. The result is expressed as a percentage. The coefficient of variation is used to compare standard deviations when the units are different for the two variables being compared. © Copyright McGraw-Hill 2004 3-25 Variance and Standard Deviation Variances and standard deviations can be used to determine the spread of the data. If the variance or standard deviation is large, the data are more dispersed. The information is useful in comparing two or more data sets to determine which is more variable. The measures of variance and standard deviation are used to determine the consistency of a variable. © Copyright McGraw-Hill 2004 3-26 Variance and Standard Deviation (cont’d.) The variance and standard deviation are used to determine the number of data values that fall within a specified interval in a distribution. The variance and standard deviation are used quite often in inferential statistics. © Copyright McGraw-Hill 2004 3-27 Chebyshev’s Theorem The proportion of values from a data set that will fall within k standard deviations of the mean will be at least 1 – 1/k2; where k is a number greater than 1. This theorem applies to any distribution regardless of its shape. © Copyright McGraw-Hill 2004 3-28 Empirical Rule for Normal Distributions The following apply to a bell-shaped distribution. Approximately 68% of the data values fall within one standard deviation of the mean. Approximately 95% of the data values fall within two standard deviations of the mean. Approximately 99.75% of the data values fall within three standard deviations of the mean. © Copyright McGraw-Hill 2004 3-29 Standard Scores A standard score or z score is used when direct comparison of raw scores is impossible. A standard score or z score for a value is obtained by subtracting the mean from the value and dividing the result by the standard deviation. © Copyright McGraw-Hill 2004 3-30 Percentiles Percentiles are position measures used in educational and health-related fields to indicate the position of an individual in a group. A percentile, P, is an integer between 1 and 99 such that the Pth percentile is a value where P % of the data values are less than or equal to the value and 100 – P % of the data values are greater than or equal to the value. © Copyright McGraw-Hill 2004 3-31 Quartiles and Deciles Quartiles divide the distribution into four groups, denoted by Q1, Q2, Q3. Note that Q1 is the same as the 25th percentile; Q2 is the same as the 50th percentile or the median; and Q3 corresponds to the 75th percentile. Deciles divide the distribution into 10 groups. They are denoted by D1, D2, …, D10. © Copyright McGraw-Hill 2004 3-32 Outliers An outlier is an extremely high or an extremely low data value when compared with the rest of the data values. Outliers can be the result of measurement or observational error. When a distribution is normal or bell-shaped, data values that are beyond three standard deviations of the mean can be considered suspected outliers. © Copyright McGraw-Hill 2004 3-33 Exploratory Data Analysis The purpose of exploratory data analysis is to examine data in order to find out what information can be discovered. For example: Are there any gaps in the data? Can any patterns be discerned? © Copyright McGraw-Hill 2004 3-34 Boxplots and Five-Number Summaries Boxplots are graphical representations of a fivenumber summary of a data set. The five specific values that make up a five-number summary are: The lowest value of data set (minimum) Q1 (or 25th percentile) The median (or 50th percentile) Q3 (or 75th percentile) The highest value of data set (maximum) © Copyright McGraw-Hill 2004 3-35 Summary Some basic ways to summarize data include measures of central tendency, measures of variation or dispersion, and measures of position. The three most commonly used measures of central tendency are the mean, median, and mode. The midrange is also used to represent an average. © Copyright McGraw-Hill 2004 3-36 Summary (cont’d.) The three most commonly used measurements of variation are the range, variance, and standard deviation. The most common measures of position are percentiles, quartiles, and deciles. Data values are distributed according to Chebyshev’s theorem and in special cases, the empirical rule. © Copyright McGraw-Hill 2004 3-37 Summary (cont’d.) The coefficient of variation is used to describe the standard deviation in relationship to the mean. These methods are commonly called traditional statistics. Other methods, such as the boxplot and fivenumber summary, are part of exploratory data analysis; they are used to examine data to see what they reveal. © Copyright McGraw-Hill 2004 3-38 Conclusions By combining all of these techniques together, the student is now able to collect, organize, summarize and present data. © Copyright McGraw-Hill 2004 3-39