Chapter 8 Descriptive Statistics

Transcription

1 8.1 Uivariate aalysis ivolves a sigle variable, for examples, the weight of all the studets i your class. Comparig two thigs, like height ad weight, is bivariate aalysis. (Which we will look at later) Data is - Data ca be split ito two categories Is the data from test scores qualitative or quatitative? Quatitative data ca be split up ito two categories: ad. - A quatitative discrete variable has exact umerical values. o Here we are workig with values of 0, 1, 2, 3, o Like the umber of sogs you have dowloaded - A quatitative cotiuous variable ca be measured ad its accuracy depeds o the accuracy of the measurig istrumet used. o Ca cotai fractios ad decimals o Legth, weight, time, etc. What is the differece betwee a populatio ad a sample? What is your defiitio of a populatio? 1

2 I statistics, the term populatio A part of the populatios is called a. - It is a subset of the populatio, a selectio of idividuals from the populatio. - Radom Samples o o Exercise 8A 8.2 Presetig data Two quick ad easy was to view data quickly ad look for patters is a ad a. Example: A studet couted how may cars passed his house i oe-miute itervals for 30 miutes. His results were: 23, 22, 22, 22, 24, 22, 21, 21, 23, 23, 27, 21, 21, 22, 23, 25, 27, 26, 23, 22, 27, 26, 25, 28, 26, 22, 20, 21, ad 20. Display the data i a frequecy table. Draw a bar chart for this data. 2

3 Whe there is a lot of data, you ca orgaize it ito groups i a grouped frequecy table. For Cotiuous data, you ca draw a histogram. It is similar to a bar chart but it does t have gaps betwee the bars. - Why are there o gaps i cotiuous data? - Oly frequecy histograms with equal class itervals will be examied. - You ca use your GDC to draw histograms. Example: The homeru totals of 76 players (ot icludig pitchers) (players played i a miimum of 40 games) of teams i the NL Cetral divisio i 2013 are: 8, 23, 7, 10, 12, 17, 6, 21, 13, 6, 6, 9, 11, 6, 1, 8, 9, 24, 18, 12, 19, 9, 21, 30, 2, 7, 0, 1, 0, 2, 18, 13, 10, 12, 12, 9, 24, 8,13,4,1,6, 4, 5, 11, 1, 4, 15, 7, 16, 5, 36, 12, 21, 5, 15, 8, 6, 3, 3, 1, 3, 12, 13, 11, 1, 9, 22, 7, 24, 5, 17, 2, 1, 0, 0 Draw a frequecy table ad histogram for the data. Exercise 8B 8.3 Measures of cetral tedecy The three most commo measures of cetral tedecy are

4 What is the mode? - What is aother ame for the mode? - Ca there be more tha oe mode? If there are two modes, what is the set called? - Whe is there o mode? Example. a. Fid the mode of 3, 4, 6, 6, 7, 7, 7, 8, 8, 9, 10, ad 10. b. Give the frequecy table fid the modal class. Weight Frequecy 110 w < w < w < w < w < Exercise 8C THE MEAN!!! The arithmetic mea is usually called the or the. The mea is the sum of the umbers divided by the umber of umbers i a set of data. Sum of the data values Mea = Number of data values The mea gives a sigle umber that idicates. - Usually ot a member of the data set - But a represetative value - The lower case Greek letter µ is the symbol for the populatio mea. o x Populatio mea µ =, where xis the sum of the data values ad N is the N umber of data values i the populatio. µ is proouced mu, (which tells us to fid the sum here) proouced sigma ad N is u. FC ask Mrs. Haley to sig the Greek alphabet sog!!!! There is ofte a misuderstadig betwee the populatio mea ad the sample mea. The populatios mea uses Greek letters whereas the sample mea uses x ad. Our course uses oly the populatio mea. 4

5 4 mea examples (they are t very ice)!!! a. Fid the mea of 32, 43, 55, 30, 62, ad 57. You ca also do this o you GDC. b. Fid the mea of the sets below. AP Score Frequecy Weight Frequecy 110 w < w < w < w < w < This is the formula as it appears i the IB Formula booklet: = i= 1 µ i= 1 f x i f i i Whe the data is grouped, we ca calculate the mea by assumig that all of the data values are equally spread aroud the midpoit. WARNING: This method leads to small iaccuracies ad that is why exam questios ofte say estimate the mea. It does ot mea guess it meas work out, as i this example or with your GDC. Last oe: Kual really likes to do well o tests. He has scores of 95, 89, 93, ad 84. What score must Kual eed to get o the fifth test i order to get a A+ score of 98% (accordig to NAFC)? Exercise 8D 5

6 The media. The media is the umber i the middle whe the umbers i a set of data are arraged i order of size. If the umber of umbers i a data set is eve, the the media is the mea of the two middle umbers. Fid the media of the 13, 11, 32, 18, 19, 20, 13, 15, 25, 29, 28, ad 20. If there are a lot of umbers ad it is difficult to fid the middle member we ca use the formula + 1 Media = th member, where is the umber of members i the set. 2 *Commo error. This formula does ot give the media. It gives the positio of the media i the data set. Exercise 8E Summary of measures of cetral tedecy 6

7 8.4 Measures of dispersio Measures of cetral tedecy (mea, media, ad mode) explore the middle of a data set. Measures of dispersio describe the spread of the data aroud a cetral value. Whe you describe a data set you should give at least of measure of cetral tedecy ad oe of dispersio. The rage is the simplest measure of dispersio to calculate. What is the rage? - It ca be effected by extreme values. - It does t tell you how the remaiig data is distributed. Quartiles: - The media of a set of data separates the data ito two halves half less tha the media, half greater. - Quartiles separate the origial set of data ito four equal parts. o Each cotais oe-quarter (25%) of the data 7

8 You ca get a sese of a data set s distributio by examiig a five statistical summary: Here is a list of combied NFL scores for two weeks of the seaso Fid the five statistical summary for the data. The differece betwee the third ad first quartile is called the iterquartile rage ( ) = Q3 Q1 A five statistical summary ca be represeted graphically as a box ad whisker plot. Draw a box ad whisker plot for the NFL data above. *Extreme or distat data values are called outliers. A outlier is ay value at least 1.5 IQR above Q3 or below Q 1. Are there ay outliers for the NFL data? Exercise 8F 8

9 8.5 Cumulative Frequecy To calculate the cumulative frequecy add up the frequecies of the data values as you go alog. A cumulative frequecy diagram (Cumulative frequecy graph) or ogive is most useful whe tryig to calculate the media, quartiles ad percetiles of a large set of grouped or cotiuous data. Usig the NFL data from the previous Draw a cumulative frequecy diagram ad media ad iterquartile rage. examples fid the Scores (s) f 15 s < s < s < s < s < s < s < 85 1 Cumulative Frequecy Make sure to label you graph properly. Exercise 8G 8.6 Variace ad Stadard Deviatio The rage ad iterquartile rage are good measures of spread but each oe is calculated from oly two data values. The combies all the values i a data set to produce a measure of spread. - It is the arithmetic mea of the squared differece betwee each value ad the mea value. If you wat to kow why there are advatages to squarig the above differece read page 276 of your book. Because the differece are squared, the uits of variace are ot the same as the uits of the data. The is the square root of the variace ad has the same uits as the data. 9

10 The formulae for the variace ad stadard deviatio are: 2 i= 1 σ = Populatio Variace = = Populatio Stadard Deviatio = ( x µ ) 2 i= 1 σ ( x µ ) 2 Example: A bag of M&Ms is supposed to weight 1.69 oz. Here is a list of 10 M&M bags: Either give studets 10 weights or do the example i the other file. Fid the mea ad stadard deviatio. O the GDC, use the value σ xfor stadard deviatio ot sx. The stadard deviatio shows how much variatio there is from the mea ad gives a idea of the shape of the distributio. - Low stadard deviatios (bottom picture) shows the data poits ted to be very close to the mea. - High stadard deviatio (top picture) idicates that the data is spread out over a large rage of values. Properties of stadard deviatio - Stadard deviatio is oly used to measure spread or dispersio aroud the mea of a data set. - Stadard deviatio is ever egative - Stadard deviatio is sesitive to outliers. A sigle outlier ca icrease the stad deviatio ad i tur, distort the represetatio of spread. - For data with approximately the same mea, the greater the spread, the greater the stadard deviatio. - If all values of a data set are the same, the stadard deviatio is zero because each value is equal to the mea. 10