ࡱ> ~} I?bjbjVV &<<iBPP4h$t:PPP: "QP@PPP+P"8P:-,m2:J|rY  0:c #"#m#m PPPPPPPPPP:PPPP#PPPPPPPPPP Y: How to Compute and Interpret Measures of Central Tendency (Mean, Median, and Mode) Objective: Learn how to compute and interpret and when to use measures of central tendency including the mean, median, and mode Keywords and Concepts Mean Median Mode Central tendency Midpoint or middle of data  EMBED Equation.3  50th percentile Most frequently occurring score Best measure of central tendency Skewed data The mean, median and mode, common measures of central tendency, represent either a typical or representative score and/or a value about which the data tend to center. A measure of central tendency frequently lies at the midpoint or middle of a data set. Mean The arithmetic average (termed the mean and abbreviated as  EMBED Equation.3 ) represents the most appropriate measure of central tendency for continuous-type data. It is obtained by adding all of the scores and dividing this sum by the number of scores Mean ( EMBED Equation.3  ) =  EMBED Equation.3  (eq. 1) where the mean can be denoted by  EMBED Equation.3  (pronounced  X-bar ) for samples and (pronounced  mu ) when dealing with populations; " denotes summation of a set of values; X represents the individual raw scores, and N equals the number of scores. Example Mean Calculations Compute the mean of the following scores placed in ascending order: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10: Step 1. Sum up the scores. "X = 1+2+3+4+5+6+7+8+9+10 = 55 Step 2. Divide "X by the total number of scores or data points (in this example 10).  EMBED Equation.3  = 55 10 (eq. 2) = 5.5 Median The median of a set of scores represents the middle value (50th percentile) when the scores are arranged as an array in order of increasing (or decreasing) magnitude. The median is often denoted by  EMBED Equation.3  (pronounced X-tilde). The median often becomes a more appropriate (representative) measure of central tendency when the data are skewed that is, the majority of scores tend to accumulate toward either the high or low end of the distribution with a few extreme scores at the opposite end. To locate the median, first rank the scores and follow these two guidelines: For an odd number of scores, the median is the middle score. For an even number of scores, the median is the mean (arithmetic average) of the two middle scores. Example Median Calculations Example #1 Compute the median for these five scores: 10, 30, 27, 29, 12. Step 1. Arrange the scores in ascending order. 10, 12, 27, 29, 30 Step 2. Because the number of total scores equals 5, an odd number, the third number (27) is the raw score at the exact middle and becomes the median  EMBED Equation.3 . Example #2 Compute the median for these six scores: 5, 6, 8, 50, 10, 70. Step 1. Arrange the scores in ascending order 5, 6, 8, 10, 50, 70 Step 2. Because the number of scores equals 6, an even number; compute the average of the values for the third and fourth middle scores. The arithmetic average of these raw scores of 8 and 10 equals the  EMBED Equation.3 , a score of 9. Mode The mode (denoted by M) represents the most frequently occurring score. When two scores occur with the same greatest frequency, each one equals the mode and the data set is considered bimodal. When more than two scores occur with the greatest frequency, the data set is said to be multimodal. To determine the mode, locate the most frequently appearing number. Example Calculations Compute the mode of the following ten scores: 10, 29, 26, 28, 15, 10, 25, 27, 10, 29: The mode, M, equals 10 Round-off Rule Carry one more decimal place than present in the original set of numbers. Round-off only the final answer, not intermediate values. For example, the mean of the numbers 2, 3, and 5 equals 3.33333,which is then rounded to 3.3. [Because the original data were whole numbers, the answer rounds to the nearest tenth.] The mean of the numbers 2.1, 3.4, and 5.7 rounds to 3.73 (one more decimal place than used for the original values). The Best Central Tendency Measure There is no single best measure of central tendency. The different measures of central tendency have different advantages and disadvantages as summarized in Table 1. Table 1. Unique aspects of mean, median, and mode.AverageDefinitionHow commonExistenceTakes every score into account?Affected by extreme scores?Advantages/ DisadvantagesMean EMBED Equation.3 ="XNMost familiar averageAlways existsYesYesTakes every score into account/ affected by extreme scoresMedianMiddle scoreCommonly usedAlways existsNoNoGood choice if there are extreme scoresModeMost frequent scoreRarely usedAlways exists\' ( ; < = > A C F wdZj h4GbEHU$j\9 h4GbPJUVmHnHujh4GbEHUjDh4GbEHUjh4GbU h4GbCJh4GbCJfHq h4GbCJH*OJQJjh4GbEHOJQJU,j  H+2  3"KfgzƛoZ(jh4GbCJEHUfHq 0jr9 h4GbCJOJPJQJUVmHnHu$jh4GbCJUfHq h4GbCJfHq j|h4GbEHU$j\9 h4GbPJUVmHnHu h4Gb56j=h4GbEHU,jK .s<$dhx$d%d&d'd-DM NOPQ]^a$ pdhx^p` $dhxa$dhx^`dhx dhx`z{|} .s1278Kҿ瑿|vqj_vWjh4GbUh4Gb5CJOJQJ h4Gb5CJ h4Gb5 h4GbCJ h4Gb6h4Gb5CJfHq (j="h4GbCJEHUfHq h4GbCJfHq  h4Gb56h4Gb$jh4GbCJUfHq (jh4GbCJEHUfHq 0j:r9 h4GbCJOJPJQJUVmHnHu  dx$If<$dhx$d%d&d'd-DM NOPQ]^a$ dhx` $dhxa$dhx#1q$dx$Ifa$$d$Ifa$ $d$Ifa$ekdd&$$IfTl4r$]$0 4 laf4T 127*$dh$Ifa$kd&$$IfTl֞4 "hTr$nF0 4 laTKLM<#<$<&<<<<<<<<<<<<<<=======?=E=E?I?Ƿdz׳׳׳׳~ѳ h4Gb6CJ h4Gb5CJhE8u0JCJmHnHuh4Gb0JCJjh4Gb0JCJU h4GbCJh4Gbjh4GbUmHnHuUh4GbCJOJQJ h4GbCJjh4GbUj'h4GbEHU,j??]Q~7|e$۷oggT!/b[o߾L&h27oVe$o&I[Vȃ> ^!H]m9}ټ:NQAt:߿88zIiZ?Ï~N !mѣG1 .7B]eYV?|v~ܳ4{{K/E,TjCWk~rn¼VjRYm⨚:3܂]әwa\ ʣ֒x<۷o?i:@]%IAY_u}͜0 /g̊āeȺ~g=ą?gg?h~׹¿3RW[ l3Ǔ-vlfN}zJvqHZKK׳+g<+ Jvŭ<>??! 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