Measures of Spread: Stadard Deviatio So far i our study of umerical measures used to describe data sets, we have focused o the mea ad the media. These measures of ceter tell us the most typical value of the data set. If we were asked to make a predictio about a member of a data set, we would use a measure of ceter to predict that value. However, measures of ceter do ot give us the complete picture. Cosider the followig test scores: Studet Test 1 Test 2 Test 3 Test 4 Johy 65 82 93 100 Will 82 86 88 84 Aa 80 99 73 88 Who is the best studet? How do you kow? Thikig about the Situatio Discuss the followig with your parter or group. Write your aswers o your ow paper. Be prepared to share your aswers with the class. What is the mea test score for each studet? 85 poits Based o the mea, who is the best studet? Aswers will vary; sice they all have the same mea, it is hard to use the mea to aswer this questio. If asked to select oe studet, who would you pick as the best studet? Explai. Some may look at Johy s icreasig tred over time. Some may look at Will s cosistecy. Some may ote Aa s oe low grade. Be sure that argumets for each studet beig the best come out, eve if you have to be the oe who makes them. The poit is, just usig the mea to describe each studet is ot eough. I thik that we ca all agree that they are ot equal i their test performace. We eed more iformatio tha just the typical test score. Oe thig to look at is how cosistet each studet is, ad measures of spread will give us that iformatio. Ivestigatio 1: Deviatio from Discuss the followig with your parter or group. Write your aswers o your ow paper. Be prepared to share your aswers with the class. Usually we calculate the mea, or average, test score to describe how a studet is doig. Johy, Will, ad Aa all have the same average. However, these three studets do ot seem to be equal i their test performace. We eed more iformatio tha just the typical test score to describe how they are doig. Oe thig we ca look at is how cosistet each studet is with their test performace. Does each studet ted to do about the same o each test, or does it vary a lot from test to test? Measures of spread will give us that iformatio. I statistics, deviatio is the amout that a sigle data value differs from the mea.
1) Complete the table below by fidig the deviatio from the mea for each test score for each studet. Score x Mea x Johy Test 1 65 85-20 Test 2 82 85-3 Test 3 93 85 8 Test 4 100 85 15 Deviatio from Score x Mea x Aa Test 1 80 85-5 Test 2 99 85 14 Test 3 73 85-12 Deviatio from Test 4 88 85 3 Score x Mea x Will Test 1 82 85-3 Test 2 86 85 1 Test 3 88 85 3 Test 4 84 85-1 Deviatio from 2) What is the sum of the deviatios from the mea? How does this relate to the mea beig the balace poit for a set of data? 0; The mea is the balace poit, so the distaces betwee the mea ad each poit balace i either directio. Ivestigatio 2: Mea Absolute Deviatio Discuss the followig with your parter or group. Write your aswers o your ow paper. Be prepared to share your aswers with the class. Oe way to measure cosistecy is to fid the average deviatio from the mea. I other words, how far do most values i a data set fall from the mea? Oe way to aswer this questio would be to fid the average deviatio, or distace, that the data values fall from the mea. So we would add up the deviatios to fid the total deviatio ad the divide by the umber of data values to fid the mea deviatio. However, the fact that the deviatios from the mea always add up to zero is a problem. No matter what we divide zero by, we always get zero! Whe talkig about spread, a value of zero idicates that there is o spread, or variability. Oe way to fix this problem is to look at oly the distaces from the mea, ad ot their directios as idicated by the sig of the deviatio (positive or egative). We ca take the absolute value of the distaces ad the fid the average distace. 1) Complete the table below by fillig i the deviatio from the mea for each test score for each studet that you calculated i Ivestigatio 1. The fid the absolute value of each deviatio.
Deviatio from Absolute Deviatio from Johy Test 1-20 20 Test 2-3 3 Test 3 8 8 Test 4 15 15 Sum 46 Will Test 1-3 3 Test 2 1 1 Test 3 3 3 Test 4-1 1 Sum 8 Test 1-5 Test 2 14 Test 3-12 Test 4 3 Sum Aa 5 14 12 3 34 2) Fid the average of the absolute deviatios from the mea for each studet. This is called Absolute Deviatio. Johy 11.5, Will 2.5, ad Aa 8.75 3) What does Absolute Deviatio (MAD) tell you about each studet? Is there oe studet who seems to be more cosistet tha the others? The MAD tells us the average deviatio from the mea for each studet, tellig us how much their test scores typically vary from their average test score. Will s MAD is the lowest, idicatig that his test scores are the least spread out. 4) Iterpret the MAD for Johy i cotext. Johy s test scores typically fall withi 11-12 poits of his average score of 85. So Johy s scores typically vary aywhere from 74 to 96.
Ivestigatio 3: Calculatig the Stadard Deviatio Below is the formula for calculatig the stadard deviatio. It looks pretty complicated, does t it? Let s break it dow step by step so we ca see how it is fidig the average deviatio from the mea. Let s start with Johy s data. ( x ) 2 Step 1: I the table below, record the data values (test scores) i the secod colum labeled Value. The x is used to deote a value from the data set. The first value is writte for you. Step 2: Fid the mea of the test scores (we did this i the Thikig About the Situatio ) ad record at the bottom of the secod colum ext to the symbol. Mu (proouced mew ) is the lowercase Greek letter that later became our letter m. is aother symbol that we use for mea (i additio to x ). Step 3: I the third colum, fid the deviatio from the mea for each test score by takig each test score ad subtractig the mea. The first differece has bee doe for you. Step 4: Add the values i the third colum to fid the sum of the deviatios from the mea. If you have doe everythig correctly so far, the sum should be zero. The capital Greek letter, called sigma, is a symbol that is used to idicate the sum. Step 5: Square each deviatio to make it positive ad record these values i the last colum of the table. The first value is doe for you. Step 6: Fid the sum of the squared deviatios by addig up the values i the fourth colum ad puttig the sum at the bottom of the colum. This is the sum of the squared deviatios from the mea. Step 7: Fid the average of the squared deviatios from the mea by dividig the sum of colum four by the umber of data values (the umber of test scores). Step 8: U-do the squarig by takig the square root. Now you have foud the stadard deviatio! The symbol for stadard deviatio is the lower-case letter sigma,. Test Value (x) Johy s Data Deviatios from Value Mea (x μ) Squared Deviatios from (Value Mea) 2 (x μ) 2 1 65 65-85 = -20 (-20) 2 = 400 2 82 82-85 = -3 (-3) 2 = 9 3 93 93-85 = 8 (8) 2 = 64 4 100 100-85 = 15 (15) 2 = 225 Mea μ = 85 Sum (x μ) = 0 Sum (x μ) 2 = 698
sum of squared deviatios (x μ)2 average of squared deviatios = = = 174.5 umber of data values sum of squared deviatios (x μ)2 σ = square root of ( ) = = 13.2 umber of data values Now repeat the process with Will ad Aa s data. Test Value (x) Will s Data Deviatios from Value Mea (x μ) Squared Deviatios from (Value Mea) 2 (x μ) 2 1 82-3 9 2 86 1 1 3 88 3 9 4 84-1 1 Mea μ = 85 Sum (x μ) = 0 Sum (x μ) 2 = 20 average of squared deviatios = sum of squared deviatios umber of data values = (x μ)2 = 5 sum of squared deviatios (x μ)2 σ = square root of ( ) = = 2.2 umber of data values Test Value (x) Aa s Data Deviatios from Value Mea (x μ) Squared Deviatios from (Value Mea) 2 (x μ) 2 1 80-5 25 2 99 14 196 3 73-12 144 4 88 3 9 Mea μ = 85 Sum (x μ) = 0 Sum (x μ) 2 = 374
sum of squared deviatios (x μ)2 average of squared deviatios = = = 93.5 umber of data values sum of squared deviatios (x μ)2 σ = square root of ( ) = = 9.7 umber of data values Discussio Questios: 1) Why is the sum of the third colum always equal to zero? The sum of the deviatios from the mea is always zero because the mea is the balace poit of the data. 2) Traslate ito words: (x μ) 2. The sum of the squared deviatios from the mea. Or from iside paretheses out, subtract the mea from each value, square these differeces ad add them up. 3) Iterpret Aa s stadard deviatio i cotext. Aa s test grades typically vary by about te poits from her average score of 85. So her test scores typically vary from about 75 to 95. 4) Who is the best studet? How do you kow? Aswers will vary. Will is the most cosistet as his scores vary by about 2.2 poits from the mea of 85 poits. Fidig the mea, media, a stadard deviatio o the calculator. Eter the followig data ito L 1 i the calculator.
***Sx is the sample stadard deviatio (divided by -1). σx is the populatio deviatio (divided by ).