Now that we know what our sample looks like, we d like to be able to describe it numerically.

Transcription

1 Descriptive Statistics: Now that we kow what our sample looks like, we d like to be able to describe it umerically. I other words, istead of havig a lot of umbers (oe for each record), we d like to be able to describe the sample with just oe or two umbers. If we oly use oe umber, what could we use? Some examples: Miimum (is this useful?) Maximum Third largest umber? Mode The umber i the middle (= media)? Mea The first two cadidates are iterestig, but really do't represet the sample. The third cadidate is just silly. The fourth (the mode) we already talked about (see p. 18 [15] {33}). It's useful to describe the data, but it's ot used much i more complicated aalyses. Oddly, almost every textbook talks about it, but the ever uses it. We'll focus o the last two, begiig with the last: I. Mea (see p. 3 & 33 [6 & 7] {41 & 4}) This measures the ceter of our distributio. I the case of a sample, it is give by: y = y i Note the symbol for the sample mea: y (remember, = sample size). This is othig ew. Everyoe kows how to calculate the average. We'll refer to this as the sample mea from here o.

2 Here is example [.15] {..3}from the book: We are give the weight (pouds) gai i six lambs over two weeks: 11, 13, 19,, 10, 1 thus we have = 56 ad we get 56/6 = 9.33 pouds. Remember, This is the SAMPLE mea. Oe ca also talk about the populatio mea or the mea of a distributio. This distictio is actually importat! We'll see why later. II. Media (p. 33 & 34 [8 & 9] {40 & 41}) The sample media is simply the value i the middle. If there is o middle umber, the it s cosidered to be halfway betwee the two middle values. I other words: If there are a odd umber of observatios, it s i the middle. If there are a eve umber of observatios, it s half way betwee the two middle values. Example (exercise..14 [.16, p. 30] {.3.3, p. 44}): Arragig the values from smallest to largest (always sort your values first!): Ad we get the media = 6.3 moles/gm (the middle value) (Ulike the mea ( y ), there is o special symbol that everyoe recogizes for the media) Example (exercise.15 [.18, p. 30] {.3.5, p. 44}): Agai, we arrage the values from smallest to largest: Now to calculate the media, we eed to take the average of the two middle umbers: = 566 = 83 Ad so the media is 83 mg/dl

3 Fially, which is better? Mea or media? (See also p. 36 [30] {43}) Depeds (do t you love a vague aswer like that?) For most thigs (particularly i this class) the (sample) mea is probably a better idicatio of the ceter. Why? Because it uses all of the data. The media uses oly the middle or middle two umbers (though the other umbers do determie where the middle is). The sample mea is extesively used i statistics, particularly the kid we re goig to lear. So why bother with the media? It does better whe the data are highly skewed, very spread out, or have lots of outliers. A commo example (ot from biology) is i icome. Listig the average icome is very misleadig. Why? Cosider Bill Gates. He pulls the average icome i his eighborhood WAY up. Also ote that icome usually does t drop below 0. The media does much better here, sice Bill Gates oly moves it up half a otch, if at all. Icidetally, just like the sample mea, what we calculated here is the sample media. So ow we have a idea of how to measure the ceter of our distributio. What about the spread? We also wat to kow: Are all the observatios sort of the same? Or are they all very differet from each other? Here is a example:

4 Here we also have some cadidates: Rage Average absolute deviatio Variace Stadard deviatio Let s go through these: I. Rage (p. 48 [p.40] {59}): Rage = maximum value - miimum value (your book also talks about iterquartile rages - igore these for ow). The ramge os sesitive to extremes (e.g. Bill Gates agai). II. So why ot use somethig like average deviatio? Here s why, usig the example from exercise.15 [.18] {.3.5} which we used above whe discussig the media: = (ote that y = ) = = = = = Problem: if we add up the differeces, we get 0. Obviously, dividig 0 by 6 is poitless, so we ca stop here. Note: the sum of the deviatios from the mea is always 0. We eed to come up with somethig else! III. Average absolute deviatios (this oe s ot i the book): Suppose we do't care if the differece is positive or egative. We just care about the total distace of each value from the mea. The we could take the absolute value of each of our umbers above.

5 So we get (remember = ): = Ad ow we have: = This (the average absolute deviatio) is used, but as it turs out, is ot terribly useful for us. The mathematics eeded to use this for doig aythig useful ca be difficult ad usually ivolve high speed computers. Icidetally, there are actually several very similar measures, but we wo't discuss them. IV. Variace (& stadard deviatio) (p [p ] {60-63}): The basic problem is that we eed to make our deviatios positive. So what else ca we do? Square the deviatios, which makes them positive, ad the take a average (well, sort of). Sample variace: Take all the deviatios ad square them. Sum these up (this, icidetally, gives you the SUM OF SQUARES (SS), a very importat quatity) Divide by -1. Usig our otatio, we have (ote the symbol for the sample variace, s ): s = y i y 1

6 Here s a example, usig the same set as above: ( ) = ( ) = 4, ( ) = ( ) = ( ) = ( ) = ( ) = ( ) = ( ) = ( ) = 1, ( ) = (7.1667) = 5, , = Sum of Squares = SS Now to get the sample variace, we divide by - 1 = 6-1 = 5: s = 11, The uits o this are (mg/dl). =, The variace is used extesively i statistics. Ofte, statisticias do t eve bother with stadard deviatios util they re ready to preset results. The problem with variace is that the uits are ot directly comparable to the origial. So we fix this ad use the stadard deviatio, s, which is simply the square root of the variace. Here s a example of the sample stadard deviatio, usig exercise..34 p. 58 [.46, p. 49] {.6.7, p. 67}: sample mea: = 44.4 ad y = 44.4/7 = sample variace: ( ) = ( ) = ( ) = ( ) = ( ) = ( ) = ( ) = = Sum of Squares = SS so the variace = s =.9571 / 6 = (remember, divide by -1; 7-1 = 6)

7 stadard deviatio: This is the square root of : s = (0.4985) = Some cocludig remarks about variaces ad stadard deviatios: Here is the formula for the stadard deviatio: y i y s = 1 The abbreviatio we use for the sample stadard deviatio is s. The sample variace is the simply s. Why o earth do we use -1 istead of i the deomiator? A ituitive explaatio (ex..31, p. 5 [p ] {6-63}): Take a sample of size 1. Now, what is the variace? Usig the formula, oe wids up with: 0 0 = udefied This makes sese, because a sample of size oe ca t tell us aythig about the variatio of a populatio. We use the sample stadard deviatio to tell us somethig about the variatio i the populatio (ot the sample). A sample of size oe has o iformatio about variability i the populatio. Also a more mathematical ote: (We'd eed at least two data poits). If you use istead of -1 that your variace will be biased (it wo't reflect the true variace). Stragely eough, the stadard deviatio is always a bit biased regardless of whether or ot you use or -1.

8 Is ever appropriate? Yes, if you re really ONLY iterested i the data you have, ad NOT i makig ifereces (estimates) about the populatio at large. This is almost ever true, ad we will pick up with this theme ext time. However, sice is occasioally (rarely!) used, please be sure you use -1 whe you calculate the variace or stadard deviatio with your calculator. Most calculators ca do it either way, so be careful! Miscellaeous stuff: sigificat digits & roudig Sice this was our first lecture with a buch of math, we should discuss sigificat digits ad roudig. I. Sigificat digits: Whe you are doe with your calculatio, you keep as may sigificat figures as the umber with the least sigificat figures you started with. For example, you have: I this case you would keep three figures i your aswer to ay calculatios where you used all three umbers. What is a sigificat figure? meas the 0 is sigificat usually meas the 0 is ot sigificat - but if this umber is part of a buch of other umbers, some of which have a umber other tha 0 i frot of the decimal, you ca assume that the 0 is sigificat. 3,000 usually meas that the last three zeros are NOT sigificat. For istace, if someoe had couted 3,03 birds durig a bird cout (yes, there are such thigs!), that perso may report it as 3,000. O the other had, if you see this umber as part of a lot of other umbers of the same magitude, but ot with three 0 s, (e.g., 3,03 or 54,806) the you ca probably agai assume the three zero s are sigificat.

9 Bottom lie: use a little commo sese, ad use the above rules as guides. Here are a few examples: a) y = NOT b) y = 6. c) y = 6 IMPORTANT: the above are the official rules o sigificat digits. May mathematicias ad scietists get very picky about them. HOWEVER, i this class, I'm ot picky at all. I'd much rather you give me too may digits tha ot eough. There's a good argumet to be made for givig more digits ad ot roudig so much (see below). II Roudig: First, ad most importat: keep all the digits your calculator or the computer ca hadle util you are all doe! Never roud aythig util you are fiished with all your calculatios! Why? because calculators ad computers ca make mistakes due to roudig errors ad other stuff. Calculators ad computers are ot ifiitely precise: Here's a simple example: /3 =?? What does your calculator say? Roudig aythig before you are doe will lead to errors! We will use the ormal rules of roudig i this class. You may occasioally hear about statistical roudig. We wo t worry about it i here (it uses a special rule if the last digit is 5 ).

10 OPTIONAL MATERIAL A easier way to calculate the variace ad stadard deviatio, ad a (gasp) proof : ** This is optioal material; we wo't cover this i class. It's a alterative method to calculate variaces that's easier if you do't have a calculator that ca do it. There's also a proof that shows the two methods (this ad the oe we leared i class) are the same. At this poit, you have bee taught the followig formula for calculatig a sample variace (ad the the sample stadard deviatio): s = y i y 1 it turs out this is equivalet to: s = y i 1 At first, you might look at that ad say yuck. But here are the details: y i The stuff i the first sum symbol is each observatio squared, ad the added up. The stuff i the secod sum symbol is all of the observatios added up ad THEN squared. You ll otice that this makes the calculatio a lot easier sice you do t have to deal with keepig track of differeces ad the squarig these. This formula was used a lot before the days of computers, but these days it's ot used much aymore. (It ca give you roudig errors if you're ot careful) Here s a example, usig exercise.34, p. 58 [.46, p. 49] {.6.7, p. 67} from your text (which we did above usig the ormal way): 6.8, 5.3, 6.0, 5.9, 6.8, 7.4, 6. Now if we square each of these we get: 46.4, 8.09, 36.00, 34.81, 46.4, 54.76, Add all these up to get 84.58

11 Now add up all the origial umbers ad get: 44.4 i.e., y i = 44.4 Now we just plug i to the formula ad we get: = Which is the same as before. Fially, here s the proof that these two formulas are the same: Note that we oly have to show that the umerator (our sum of squares )is the same, sice our deomiator is already the same: i =1 y i y = y i y y i y expadig the square = y = = y i y y y i =1 i =1 y i y i y rearragig makig use of the fact that y = y i = y i y y i cacellig out y i