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1 LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest" fuctioality; through the fuctio directly ad through the "aalysis tools" set of macros. These istructios cover usig "liest" as a spreadsheet fuctio. Usig liest i your spreadsheets is very easy, after you master the cocept of a array fuctio. Array fuctios are fuctios that while etered ito a sigle spreadsheet cell produce results that fill several cells. The steps outlied below take you setbystep through the process of liear curve fittig. Step 1. Type i your data i two colums, oe for the x variables ad oe for the y. You ca use ay labels you would like. "x" ad "y" are used i the example at right for coveiece. Step 2. Select the area that will hold the output of the array formula. For "liest" you should drag to form a 5 row by 2 colum data array. Step 3. Click i the formula bar at the top of the scree. Now press the fuctio wizard butto. This butto is i the formula bar ad is labeled "fx". A twopart scroll box will appear; i the left scroll box click o "Statistical" ad i the right click o "LINEST". Next click o "Next>." The widow show below will appear. O your spreadsheet select the cells cotaiig the y values by
2 draggig i the origial spreadsheet usig the mouse. Click i the "kow_x's" dialog box, ad select the cells cotaig the x values. Type i "TRUE" i the last two dialog boxes. The first TRUE idicates that you wish the lie to be i the form y=mx+b with a ozero itercept. The secod TRUE specifies that you wish the error estimates to be listed. The Fuctio Wizard dialog box should the appear as below. Step 4. Click o "Fiish." The formula bar should the appear as below, although your y ad x cell rages may be differet, of course. If the values are icorrect, you ca edit them as you would ormally. Step 5. Now here is the importat step. LINEST is a array fuctio, which meas that whe you eter the formula i oe cell, multiple cells will be used for the output of the fuctio. To specify that LINEST is a array fuctio do the followig. Highlight the etire formula, icludig the "=" sig, as show above. O the Macitosh, ext, hold dow the apple key ad press "retur." O the PC hold dow the Ctrl ad Shift keys ad press Eter. Excel adds "{ }" brackets aroud the formula, to show that it is a array. Note that you caot type i the "{ }" characters yourself; if you do Excel will treat the cell cotets as characters ad ot a formula. Highlightig the full formula ad typig the apple key or Ctrl + Shift ad "retur" is the oly way to eter a array formula. The least squares results should be prited as show below. The labels i the first ad last colum are't provided by the LINEST fuctio. We've added them to show the meaig of each cell. For example, the slope is 2.629±0.085 ad the itercept is ± 0.41.
3 x y slope itercept ± ± r s(y) F degrees of freedom regressio ss residual ss Step 6: You should ow evaluate the model that you have built. The r 2 value is ofte used for this purpose, but it is oly a rough idicator of the goodess of fit. The r 2 value is calculated from the total sum of squares, which is the sum of the squared deviatios of the origial data from the mea: total ss = (yi yav) 2 ) ad the regressio sum of squares, which is the sum of the squared deviatios of the fit values from the mea: regressio ss = (y^i y av ) 2 Givig: r 2 = regressio ss total ss = (y^ y av ) 2 ( y i y av ) 2 Values close to oe are good. The ucertaities i the slope ad itercept are much better for judgig the quality of the fit. I the example the ucertaity i the slope is 0.085/2.629*100 = 3% ad the ucertaity i the itercept is 12%, which is oly about two sigificat figures i each. The ucertaities i the slope ad itercept are ot as good as the r 2 of might have idicated! A eve better statistical test of the goodess of fit is to use the Fisher Fstatistic. The Fstatistic is the ratio of the variace i the data explaied by the liear model divided by the variace uexplaied by the model. The Fstatistic is calculated from the regressio sum of squares ad the residual sum of squares. The residual sum of squares is the sum of the squared residuals: residual ss = (y i y^i) 2 = r 2 i Dividig by the degrees of freedom, gives the variace of the y values
4 r 2 i s 2 y = 2 The regressio sum of squares, the residual sum of squares, ad the stadard deviatio of the y values, s(y) are all listed i the liest output. The Fstatistic is the the ratio of the variaces: F= variace explaied variace uexplaied = regressio ss/v ( (y^i y av ) 2 ) 1 /v1 residual ss/v = 2 ( (y i y^i) 2 ) /v2 You use the Fstatistic uder the ull hypothesis that the data is a radom scatter of poits with zero slope. Critical values of the F statistic are listed i stadard statistics texts, the CRC Hadbook, ad Quatitative Aalysis texts. If the Fstatistic is greater tha the Fcritical value, the ull hypothesis fails ad the liear model is sigificat. For the degrees of freedom, which are abbreviated i most tables as v 1 ad v 2,usev 1 =1adv 2 =k,wherekis the umber of variables i the regressio aalysis icludig the itercept ad is the umber of data poits. The value for v 2 is listed as the degrees of freedom i the liest output. A small part of the Ftable is show at right for a α value of 0.05, that is, 95% cofidece. For the example above, v 1 =1adv 2 =6 2=4.The Fcritical value is The Fstatistic for our example is , which is much greater tha the Fcritical value. You are 95% sure that your data is ot a radom scatter of poits ad that the regressio is justified. Fcritical values at α=0.05 v 2 F(v 1 =1) Step 7. You will ow eed to calculate the fit y values, y^ i, which are the values that lie o the lie at the give x values. You ca use the TREND array fuctio for this, but it is just as easy to simply calculate the fit y values directly. Start a ew colum ext to the y values. I this ew colum eter the formula that gives y^ i =mx i + b, with the slope ad itercept from the LINEST output:
5 Step 8. You ca ow use the "Chart Wizard" to help graph the results: first select the three colums i your spreadsheet. Iclude the colum labels. Click o the Chart Wizard ico: The cursor will chage shape idicatig that you are to drag o your spreadsheet where you wat the plot to appear. Remember for lab reports that charts should be at least half a page. The Wizard will the take you through settig up your graph. Do a scatter graph, ad choose the format that has plot symbols, but ot coectig lies. Step 9. You ow eed to replace the plottig symbols for the fit y values poits with coectig lies. Double click o oe of the fit y value data poits. The "Format Data Series" dialog box will appear. Chage the default settigs to o plot symbol (marker) ad coectig lies as show below: The plot should ow appear as at right. Charts ad spreadsheet cells ca easily be copied ad pasted ito Word documets LINEST Tutorial Plot y fit y x (uits?)
6 Addig Error Bars to Plots After you have your graph displayed you ca easily add error bars. Double click o oe of the plottig symbols for your data. The dialog box show below will appear. Click o the "Y Error Bars" tab. Click o the "Both" ico. Next click o the "Custom" butto. Next click i the "+" box ad the select the cell i your spreadsheet that cotais the s(y) value. Repeat this last step i the "" box. Click o "OK" ad the error bars should appear o your plot. The fial chart, i all its glory looks like this: 16 LINEST Tutorial Plot y fit y x (uits?)
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