Statistical Analysis and Graphing

Transcription

1 BIOL 202 LAB 4 Statistical Aalysis ad Graphig Aalyzig data objectively to determie if sets of data differ ad the to preset data to a audiece succictly ad clearly is a major focus of sciece. We eed a way to compare data sets that does ot rely o our bias as to what we thik is happeig. We use statistics to make objective decisios about differeces. Scietists create graphs ad tables to cocisely report data results. Comparig Two Meas (95% Cofidece Itervals) Istructios follow for calculatig a approximatio of the 95% cofidece iterval for a sample mea (95% CI). What is the 95% CI? It represets a predictio of the true value for the mea of the uiverse from which you have take a sample. That is, if you were to repeat your measuremets agai o aother similar sample, you would expect to fid a slightly differet mea. But you would also expect your secod sample to have a mea that is fairly close to your first determiatio. How close? It is possible to use your origial data to predict, with 95% cofidece, the iterval withi which the true mea lies. How is this useful? Cosider the ext situatio. If you were to take aother sample ad it s mea was NOT withi your predicted iterval, the you would suspect that your secod sample was really a very differet thig from the first oe. I fact, with kowledge of the 95% cofidece itervals you may say whether two meas are sigificatly differet, or ot sigificatly differet. This is very importat. If the meas are ot sigificatly differet (eve though they are differet umbers), the you coclude that the two samples (represeted by their meas) are really the same i spite of the slightly differet umerical values for the meas. If the meas are NOT sigificatly differet, it meas that basically the differeces you have oted are just a fluke that happes wheever you measure the same thig more tha oce Coversely, if two meas are sigificatly differet, it meas that the samples are really from differet uiverses. Some thigs to ote: 1) The larger your sample size, the closer the short-cut method will come to the real thig--the real statistical determiatio. A sample size of ie or te is about the smallest that is reasoable to use. 2) The purpose of all of these calculatios is to arrive at the 95% CI. To get that, however, you eed the stadard error (SE). But i order to get SE, you eed the stadard deviatio (s). Ad to get s you eed the mea (the average, x _ ). Therefore, the steps described below begi with how to calculate the mea ad build to gettig the 95% CI. 1

2 Here are the steps ad a idetificatio of symbols: 1) Compute the mea or average for each sample (group of measuremets). As you kow the average is obtaied by summig all measuremet values ad dividig this sum by the umber of measuremets. This is writte symbolically as follows. Here x _ x _ = Xi i=1 umerator i=1 (Eq. 1) is the symbol typically used for the mea or average of a sample, ad the Xi simply idicates summig the idividual measuremets ( each measuremet is a X, e.g. X 1,, X 2, X 3, etc.) from the first value (i =1 would be the first X, or X 1 ) to the last value (i = ). The symbol is always used to idicate the sample size or umber of available measuremets. 2) Use a calculator to get the stadard deviatio, s. May calculators yield the mea ad stadard deviatio with a sigle keystroke whe the data are etered usig a special key (usually idicated with a summatio symbol, ). A descriptio of the appropriate steps ad keys to use will be i the istructio maual for your calculator. Your istructor may be able to help decipher those istructios so brig them to class. You do ot eed to kow it, but here is the formula describig what the calculator is doig. IF your calculator does ot have this fuctio, it is surprisigly easy, though time cosumig, to do this calculatio yourself. The stadard deviatio, s, may be computed as s = (X i - x _ ) 2 i=1-1 (Eq. 2) The umerator [ (X i - x _ ) 2 ] simply idicates takig the differece betwee the i=1 mea ad each idividual measuremet, squarig this differece, ad summig the squared differeces for all the measuremets. The sum (of the squares) is the divided by the sample size less oe, ad the square root is take of the remaiig value. 3) Calculate the stadard error of the mea, SE. It is computed as follows 2

3 SE = s (Eq. 3) This calculatio is easy. Simply divide the stadard deviatio, step 2, by the square root of the sample size,. 4) Calculate the 95% cofidece iterval of the mea. (This is the iterval withi which the actual mea for the populatio is predicted to fall.) The cofidece that the mea is withi this iterval is 95%, hece the ame. To get the iterval you must calculate a factor to add to the mea to get the top of the iterval, ad subtract the factor from the mea to get the bottom of the iterval. To calculate (approximately) the factor to add ad subtract from the sample mea, simply multiply the stadard error by two. That is, the factor is (2)(SE). The 95% cofidece iterval is the mea ± the factor, as described by equatio 4 95% CI = x _ ± (2)(SE) (Eq. 4) Two umbers (or limits) describe the iterval. There is a lower umber, or lower limit (x _ - 2 SE), ad a upper limit (x _ + 2 SE). The lower 95% cofidece limit is called 95% LCL, ad the upper limit is 95% UCL. These are oted: 95% LCL = x _ - 95% CI factor (Eq. 5) 95% UCL = x _ + 95% CI factor (Eq. 6) Now it is your tur to try the calculatios. A sample data set is provided below. Sample data: Domiat No-domiat Calculate the mea, stadard deviatio, stadard error of the mea, 95% cofidece iterval factor ad 95% cofidece limits. The correct values (called the reduced data ) are foud also i Table 1. The reduced data preseted i Table 2.1 ca be iterpreted very quickly whe placed i graphical form as i Figure 1. Each rectagle represets the 95% cofidece iterval (The top of the box is 95% UCL, the bottom of the box is 95% LCL). The horizotal lie idicates the mea, ad the log vertical lie idicates the overall rage of observatios (the vertical rage lie coects the largest ad smallest idividual measuremets). By the way, to give proper credit to the scietists who suggested displayig reduced data i this fashio, this type of graph is called a Dice-Leraas graph. 3

4 Table 2.1. Idividual measuremets (= raw data) ad reduced data for maximum grip stregth i the domiat ad o-domiat hads of ie studets. 95% cofidece iterval mea s SE CI factor LCL UCL Domiat No-domiat Maximum Stregth (kg) No-domiat Domiat Figure 2.1. Maximum stregth of the o-domiat ad domiat had i ie studets. Figure 2.1 allows for a easy ad quick decisio to be made as to whether the grip stregths of the domiat ad o-domiat hads are statistically similar or differet. I this case the decisio is that the stregth of the domiat had is statistically differet from the stregth of the o-domiat had. The reaso for this coclusio is that the rectagles idicatig the 95% cofidece itervals clearly do ot overlap oe aother. Figure 2.2, below, is similar to Figure 2.1 except that some additioal distributios have bee added so that you may practice decisio-makig regardig sigificace of differece betwee meas. 4

5 35 Maximum Stregth (kg) A B C D E 20 Figure 2.2. Maximum grip stregth of the o-domiat (A) ad domiat had (C) i ie studets. Bars B, D, ad E are hypothetical ad have the 95% cofidece iterval of the domiat had but a differet mea. The basic rules for comparig these sorts of graphical represetatios are very simple. There is a statistical differece whe the meas of two samples do ot fall withi oe aother s 95% cofidece iterval. There is o statistical differece whe the mea of oe sample falls iside the 95% cofidece itervals of the sample to which it is beig compared. Your tur. Compare all the meas two at a time. What do you coclude? Are the meas sigificatly differet, or is there o sigificat differece? (No sigificat differece idicates that the meas are actually the same i spite of their umerical differece.) Check below for the correct coclusio. Compariso A vs. C A vs. B A vs. E itervals A vs. D. Coclusio/(reasoig) there is a statistical differece betwee A ad C (o overlap betwee the respective 95% cofidece itervals) there is a statistical differece betwee the two sets of data (o overlap betwee the respective 95% cofidece itervals) there is o statistical differece betwee the two meas (because: a) there is overlap betwee the respective 95% cofidece ad b) the mea of A falls i the 95% cofidece iterval of E ad c) the mea of E falls i the 95% cofidece iterval of A.) there is a statistical differece (although the 95% cofidece itervals overlap 5

6 a) the mea of A does ot fall withi the 95% cofidece iterval of D, ad b) the mea of D does ot fall withi the 95% cofidece iterval of A. Data to Aalyze: You will prepare a Results sectio for the data below. (15 Poits) For the followig data, compute the mea, stadard error, ad 95% C.I, prepare a Table for the reduced data ad plot these data o a graph (computer-geerated see istructios below). Determie which groups, if ay, statistically differ from each other. These data are the sout-vet legth (i mm) of blut-osed leopard lizards, a edagered species, which occurs here i the Sa Joaqui Valley. Figure 2.3. Male blut-osed leopard lizard (Gambelia sila) from the Sa Joaqui Valley, Califoria. Sout-Vet Legths (mm) of Blut-Nosed Leopard Lizards: Males: 103, 115, 112, 113, 108, 109, 120, 116, 117, 109, 111, 113, 110, 119, 117 Females: 110, 109, 115, 106, 114, 113, 118, 107, 108, 115, 114, 112, 110, 115 Juveiles: 85, 93, 76, 56, 92, 88, 75, 78, 82, 85, 79, 81, 67, 77 6

7 Istructios for producig a error bar graph i Microsoft Excel Step 1. Ope a worksheet i MS EXCEL ad eter the data i the followig format: Treatmet 0.1 M 0.2 M 0.1 M M.43 Note: it is crucial that the meas be i either the same row or colum! The first colum cotais the labels for the two treatmets. The secod colum cotais the mea for treatmet 1 (0.1M). The third colum cotais the mea for treatmet 2 (0.2M). Step 2. Highlight the data, icludig the labels i the treatmet colum. Step 3. Click o the chart Wizard ico o the toolbar (it is the small bar graph). Or you ca click o (isert/chart) i the meus Step 4. Select uder chart type Lie from the list of chart types ad select the subtype with markers displayed at each data value (this is usually the secod choice i the first colum for Excel). The click ext. Step 5. Uder Data Rage click series i: Colums Step 6. Click the Series tab 7

8 Step 7. Check the followig settigs uder the Series widow. If everythig worked right, you should t have to chage aythig, but you ca use these steps to fix ay problems. A. Category (x) axis labels Click o the red arrow by the category axis labels; whe the excel sheet is visible highlight the category colum (the first colum with the treatmet ames ad oly highlight the ames, ot the title of the colum) B. Series labels: These should be series 1 ad series 2. If they are there, go o to the ext step. If there are o series listed you will have to add the series (meas of treatmet 1 ad treatmet 2). Click o the add butto, the for series1, click o the red arrow i the values category. This will take you to the worksheet. Highlight the colum with the first mea ad press retur (or click o the red arrow). Repeat this process for the 2 d mea. After settig both the series ad category labels click ext. Step 8. Click o each of the tabs to display the available optios Titles: a) Do ot give the chart a title b) Type i the label for the X-axis title: this is the idepedet variable Acid cocetratio (M) c) Type i the label for the Y-axis title: this is the depedet variable- Chage i mass (g) Gridlies: tur off gridlies by clickig o each checkmark util oe are displayed. Leged: Tur off the leged by clearig the show leged box (o checkmark displayed) Tabs Step 9. Click ext ad the fiish. 8

9 Step 10. You will ow have a chart with a marker i a colum for each idepedet variable category. Drag the cursor over oe of the markers util a label pops up over the marker, the double click with the mouse. This will ope a format data series box. Select the Y Error Bars tab Uder Display click o the Both optio the click fixed value. eter the value of 2*SE i the box to the right. Select the patters tab; uder marker, select style ad chage the marker to a horizotal lie. You ca also chage its size ad colors i this meu. To exit click ok. Step11. Repeat step 10 for each idepedet category (each marker should ow have error bars) Step 12. Other editig: You ca edit most of the thigs i your graph by double clickig o the item. This will brig up a meu which you ca use to make chages to the graph. The most frequet such chage you may make is adjustig the x or y scale (double-click o the appropriate axis to brig up the meu). Remember that there is o eed to start a axis at zero, particularly if doig so squeezes all of your data ito oe part of the graph. Play with differet high ad low values util you get a graph with the data icely spread (but do t cut off your error bars!). Oce you are happy with the appearace, go to step 13. Step 13. Select the chart ad uder the edit commad; click copy. Step 14. Ope MS Word ad ope a ew documet. Step 15. Oce the ew documet is ope eter retur twice ad select edit ----paste commad. Step 16. Your chart is ow i MS word ad you ca add the Figure Captio. Chage i Mass (g) Acid Cocetratio (M) Figure 1. The mea weight chage (grams) i atacid tablets exposed for te miutes to oe of two differet acid cocetratios. N=10 atacids per cocetratio. The meas are represeted by the cetral horizotal lies. The error bars represet the 95% cofidece iterval. 9