Statistical Methods. Chapter 1: Overview and Descriptive Statistics

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1 Geeral Itroductio Statistical Methods Chapter 1: Overview ad Descriptive Statistics Statistics studies data, populatio, ad samples. Descriptive Statistics vs Iferetial Statistics. Descriptive Statistics Pictorial ad tabular methods Stemplot, dotplot, histogram, boxplot. Numerical measures Measures of Locatio: Mea ad Media. Measures of Variability: Rage, Variace, ad IQR. Iferetial Statistics Draw coclusios about a certai populatio parameter. Cofidece Itervals. Hypothesis Testig. What does statistics study? Statistics is a mathematical sciece pertaiig collectio, presetatio, aalysis ad iterpretatio of data. Populatio: a well-defied collectio of objects. Sample: a subset of the populatio. Variable: characteristics of the objects. Observatio: a observed value of a variable. Data: a collectio of observatios. statistics study data uderstad the populatio About Variable What is variable? Characteristics of a populatio of iterest whose values vary. A variable ca be Categorical e.g. x = geder of a perso (male, female) Numerical Discrete variable: e.g. x = # of studets i a class Cotiuous variable: e.g. x = height of a studet

2 Types of Data Data come from makig observatios either o a sigle variable or simultaeously o two or more variables. Uivariate data: observatios o a sigle variable Bivariate data: observatios o two variables e.g. (x, y) =(height, weight) of a studet Multivariate data: observatios o more tha two variables e.g. (x, y, z) = (height, weight, geder) of a studet How to study data? What is Statistics? Data collectio Samplig methods, experimetal desig. Data aalysis, presetatio & iterpretatio Descriptive statistics - summarize ad describe features of data Visual methods: dotplot, pie chart, histogram. Numerical methods: measures of locatio ( mea, media) ad variatio (rage, variace) Iferetial statistics - make iferece about the populatio from samples Poit estimate, cofidece itervals, hypothesis testig. Iferetial Statistics ad Probability Theory

3 Descriptive Statistics: Visual Methods Stem-ad-leaf display Dotplot Histogram Boxplot Stem-ad-leaf Display Example 1 The umber of touchdow passes throw by each of the 31 teams i the Natioal Football League i 000 is give below: {14, 9,, 18, 0, 15, 6, 9, 3, 18, 19, 18, 3, 8, 37, 1, 14, 19, 1, 0, 16,, 33, 8, 1, 18,, 14, 33, 1, 1} What does the data tell? The tes digits called stems are arraged as a colum to the left. The oes digits are listed to the right of each stem ad are called leaves What ca we say about the data set ow?most teams had 10 9 touchdow passes. Refied Stem-ad-leaf Display Whe too may leaves are lumped ito a few stems, splittig the stem helps reveal more iformatio about the distributio of data. We ca further refie the above stem-ad-leaf display by splittig each stem ito two parts: low ad high. 0H 69 1L 444 1H L

4 H 988 3L 33 3H 7 What ca we say about the data set ow? Most teams had 15 4 touchdow passes. Compare Data by Stem-ad-leaf Display Example Suppose we also have data from the 1998 seaso. We ca compare the umbers of touchdow passes i the 1998 ad 000. Year 1998 Year H L H L H L 33 3H L The peaks of the two seasos are slightly differet. For both seasos, most teams had 15 4 touchdow passes. The shapes of the data distributios are similar. Summary: Stem-ad-leaf Display How to make a stem-ad-leaf display? 1. Select oe or more leadig digits for the stem values (ay value appropriate). The trailig digits become the leaves.. List possible stem values i a vertical colum. 3. Put the leaf for each observatio besides the correspodig stem. 4. Idicate the uits for stems ad leaves. What ca a stem-ad-leaf display tell? Typical value Symmetry of distributio Peaks Outliers Stem-ad-leaf display is suitable for a data set with a moderate size. Dotplot Example 3 O-rig temperatures (F ) for test firigs or actual lauches of the shuttle rocket egie. {84, 49, 61, 40, 83, 67, 45, 66, 70, 69, 80, 58, 68, 60, 67, 7, 73, 70, 57, 63, 70, 78, 5, 67, 53, 67, 75, 61, 70, 81, 76, 79, 75, 76, 58, 31} 4

5 Dotplot of the O-rig temperature data Temperature Summary: Dotplot How to make a dotplot? 1. Represet each obs by a dot above the correspodig locatio o a measuremet scale.. Stack dots vertically whe a value occurs more tha oce. What ca a dotplot tell? Locatio of typically values Spread of data set Extreme values Gaps betwee values Dotplot is a ice display of data whe a data set is reasoably small or has oly a few distict values. Histogram What if a data set is large? Use Histogram For differet types of data, we costruct histograms differetly. Histogram for discrete data Histogram for cotiuous data Histogram for categorical (qualitative) data, also kow as Bar-graph Histogram for Discrete Data Frequecy (Cout) I a discrete data set, frequecy of a value c is the umber of occurreces of c i the data set. Relative frequecy The relative frequecy of a value c is frequecy of c relative frequecy of a value c = where is the total umber of observatios i the data set. If we list frequecies of a data set i a table, it is called frequecy distributio/table. 5

6 Costructig Histogram for Discrete Data How to create a histogram for a discrete data set? 1. Determie the distict values c 1, c, c 3,..., c r i the data set.. Calculate the relative frequecy for each c j, j = 1,,..., r: relative frequecy of c j = umber of occurreces of c j 3. Mark the c j s o a horizotal scale, draw a rectagle whose height is the relative frequecy of c j, where (j = 1,,..., r). The area of the rectagle is proportioal to the relative frequecy. Histogram for Discrete Data Example married couples betwee 30 ad 40 years of age are studied to see how may childre each couple have. Table below is the frequecy table of this data set Kids # of couples Relative Freq Histogram of Example 4 Histogram for Cotiuous Data How to create a histogram for a cotiuous data set? 1. Divide the measuremet axis ito a umber of class itervals/classes such that each obs falls ito exactly oe iterval. Deote these itervals by: I 1, I,..., I r. To esure that each obs falls ito exactly oe iterval, we may use itervals i the form: I 1 = [a 1, a ), I = [a, a 3),... We may use I j s of the same iterval legth, this is called equal class width; we may also use I j s of differet iterval legths, this is called uequal class width j = 1,, 3,..., r.. Calculate relative frequecy for each iterval I j, j = 1,, 3,..., r. 3. Draw a rectagle above each I j. For equal class width case, rectagle height = relative frequecy. For uequal class width case: rectagle height = relative frequecy of the class iterval I j class iterval width, the resultig rectagle heights here are called desities. The area of the rectagle is proportioal to the relative frequecy. For uequal class width histograms, the total area of all rectagles is 1. 6

7 Histogram for Cotiuous Data Example 5 Adjusted eergy cosumptio durig a particular period for a sample of 90 gas-heated homes are recorded. We divide the class itervals as follows: Class [1, 3) [3, 5) [5, 7) [7, 9) [9, 11) [11, 13) [13, 15) [15, 17) [17, 19) Freq Relative freq Histogram of Example 5 Histogram Shapes Histograms have a variety of shapes, the shape of a histogram coveys importat iformatio about the distributio of data. Uimodal: Sigle peak Bimodal: Two peaks Multimodal: Two more peaks Symmetric: Left right Positively skewed: Right tail stretchig out Negatively skewed: Left tail stretchig out 7

8 Histogram Shapes Descriptive Statistics: Numerical Measures Visual displays give us geeral ideas about the shape of data distributio, typical values. Numerical measures give us quatitative measures istead. Measures of locatio Mea Media Trimmed mea Quartiles Measures of variability 8

9 Variace Stadard deviatio Aother visual display of data: Boxplot. Measure of Locatio: Mea Sample mea of a sample of size {x 1, x,..., x } is the arithmetic mea of all obs i the data set ad is deoted by x: i=1 x = x i Iterpretatio of x: measures locatio/ceter of a sample. x takes every idividual obs ito accout ad weigh them equally. Populatio mea is average/ceter poit of a populatio, ad is usually deoted by µ. Use sample mea x to estimate ad make ifereces about the usually ukow populatio mea µ. Sample Mea Example 6 The followig sample cotais weights (lbs) of basses i a specific lake: {x 1 = 1., x = 1.51, x 3 = 1.34, x 4 = 1.60, x 5 = 0.98, x 6 = 1.71, x 7 = 1.8, x 8 = 1.04, x 9 = 1.10, x 10 = 0.85, x 11 = 1.08} The mea weight of this sample is: x = = Suppose we catch aother bass i the lake ad it weighs 1.5 lbs. {x 1 = 1., x = 1.51, x 3 = 1.34, x 4 = 1.60, x 5 = 0.98, x 6 = 1.71, x 7 = 1.8, x 8 = 1.04, x 9 = 1.10, x 10 = 0.85, x 11 = 1.08, x 1 = 1.5} The mea weight of this sample becomes: x = =.3 Drawback: Sample mea is very sesitive to outliers. Alterative measure: Media Measure of Locatio: Media Sample media of a sample of size {x 1, x,..., x } is the middle value of the sample, deoted by x. It is obtaied by: 1. Order the obs from smallest to largest {x (1), x (),..., x () }.. The media is the: x ( +1 ) whe is odd x = x ( ) +x ( +1) whe is eve Iterpretatio of x : the value i the middle of the sample Note that to calculate sample media, oly oe or two obs i the middle are eeded. Populatio media is the middle poit i a populatio, ad is usually deoted by µ. Use sample media x to estimate ad make ifereces about the usually ukow populatio media µ. 9

10 Sample Media - Example 6 Before we caught the huge bass, we had = 11 obs i the sample: 1. Order the data set from smallest to largest: x (1) = 0.85, x () = 0.98, x (3) = 1.04,..., x (6) = 1.,..., x (11) = 1.8. is odd, so x = x ( 11+1 ) = x (6) = 1. Comparig x = 1.30 ad x = 1., the differece is ot big. Now after we caught the 1.5-lb fish, our sample size becomes = 1, ad media: 1. Order the data set from smallest to largest: x (1) = 0.85, x () = 0.98, x (3) = 1.04,..., x (6) = 1., x (7) = 1.34,..., x (11) = 1.8, x (1) = 1.5. is eve, so x = x (6)+x (7) = = 1.8 Media is clearly ot severely affected. Measures of Locatio: Trimmed Mea x is sesitive to outliers, while x is very isesitive to outliers, two extremes. A trimmed mea is a compromise betwee these two. Give the umber α, where 0 < α < 1, the 100α% trimmed mea is computed by elimiatig the smallest ad largest 100α% i the sample ad the calculate the average over the obs left i the sample. See details i your textbook (page 8). Measures of Locatio: Quartiles Media separates the sample ito two parts: lower sub-sample ad upper sub-sample. odd: {x (1),..., x ( +1 )} ad {{x ( +1 )},..., x ()} eve: {x (1),..., x ( ) } ad {{x ( +1) },..., x () } Quartiles divide the lower ad upper sub-samples ito two parts: 1st Quartile: Q 1 = media of the lower sub-sample, also called the lower fourth d Quartile: Q = media of the etire sample 3rd Quartile: Q 3 = media of the upper sub-sample, also called the upper fourth Iter Quartile Rage: IQR = Q 3 Q 1, also called fourth spread Quartile Example Still use our bass example, rak the 11 obs: {x (1) = 0.85, x () = 0.98, x (3) = 1.04, x (4) = 1.08, x (5) = 1.10, x (6) = 1., x (7) = 1.34, x (8) = 1.51, x (9) = 1.60, x (10) = 1.71, x (11) = 1.8} Q 1 = x (3) + x (4) = Q = x = 1. Q 3 = x (8) + x (9) = IQR = =

11 Measures of Variability Data set 1 { 0.0, 0.10, 0.01, 0, 0.01, 0.10, 0.0}, Sample mea: x 1 = 0 Data set { 10000, 000, 100, 0, 100, 000, 10000}, Sample mea: x = 0 Two data sets have the same meas, but obviously secod oe is more spread out. So we eed umeric measures of such variability too. Variace is oe of such measures. Measures of Variability: Variace To compute sample variace for a sample {x 1, x,..., x } 1. calculate the sample mea x. calculate the deviatios of each obs from x: x 1 x, x x,..., x x 3. s is the average sum of squares of the deviatios: s = i=1 (x i x) 1 iterpretatio: average magitude of the deviatio from the sample mea Sometimes we also use sample stadard deviatio: s = s Similarly, we also have populatio variace σ ad populatio std dev σ as a measure of variability of the populatio. s /s could be used to estimate or make ifereces about σ /σ. The Divisor 1 Why do we use 1 as the divisor to calculate s? We hope s ca be a good estimate of σ, ideally, we wat to calculate s as: s = (xi µ) µ is somethig ukow from the populatio, a replacemet of µ is x, but obs i a sample ted to be closer to the sample mea x, resultig a relatively smaller sum of squares, so we use 1 istead of as the divisor to compesate for this. 1 is called degree of freedom. This is because s is based o deviatios x 1 x,..., x x, but sice (x i x) = 0, ay 1 deviatios will be eough. Properties of s A workig formula for s s = Sxx 1, S xx = x i ( x i ) Properties of s Let {x 1, x,..., x } be the sample ad c be ay ozero costat. If y 1 = x 1 + c, y = x + c,..., y = x + c, the s y = s x. If y 1 = cx 1, y = cx,..., y = cx, the s y = c s x ad s y = c s x. 11

12 Sample Variace Let us look at the data sets that have the same mea. Data set 1: { 0.0, 0.10, 0.01, 0, 0.01, 0.10, 0.0}, x 1 = 0 s 1 = = Data set : { 10000, 000, 100, 0, 100, 000, 10000}, x = 0 s = = Boxplot Boxplot is very useful i describig several of a data set s importat features such as: ceter, spread, symmetry ad outliers. 1. Draw a horizotal axis, fid Q 1, Q ad Q 3 ad calculate IQR.. Place a rectagle above the axis, with the left edge at Q 1, right edge at Q Place a vertical lie segmet iside the rectagle at the locatio of Q. 4. Draw whiskers out from each ed of the rectagle to the smallest ad largest obs. Boxplot With Outliers We ca also draw boxplots that show outliers. Ay obs farther tha 1.5IQR from the earest quartile is a mild outlier Ay obs farther tha 3IQR from the earest quartile is a extreme outlier To draw boxplot that show outliers, we modify the boxplot by: 1. Drawig a whisker out from the rectagle to the smallest ad largest obs that are ot outliers.. Plot mild outliers by solid dots, plot extreme outliers with circles. (optioal) Boxplot Example 7 This is example 1.18 i your textbook (page 37) Pulse width data, = 5: We have: {5.30, 8.0, 13.80, 74.10, 85.30, 88.00, 90.0, 91.50, 9.40, 9.90, 93.60, 94.30, 94.80, 94.90, 95.50, 95.80, 95.90, 96.60, 96.70, 98.10, 99.00, , , , } So, the extreme outliers are: The mild outliers are: Q 1 = 90., Q = x = 94.8, Q 3 = 96.7, IQR = IQR = 9.75, 3IQR = , 8, 0, ,

13 Pulse Width Boxplot of Example 7 Distributio Shapes, Boxplots ad Measures of Locatio 13