ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics

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1 ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the oe thig we might ask ourselves is what is their average. However there are several differet measures of average ad i this sectio we will look at the three most commo. The first is perhaps what most people thik of whe they say average ad its defiitio is as follows. Defiitio Arithmetic Mea. Give a list of umbers x 1,x,...,x which do t have to be distict, the their arithmetic mea is defied to be x = 1 x 1 +x + +x = 1. Remark This particular mea is called the arithmetic mea to distiguish it from several other meas. Perhaps the most commo of these is called the geometric mea. To calculate the geometric mea we multiply the umbers ad the take the th root, rather tha addig the umbers ad the dividig by. We will oly deal with the arithmetic mea i this course however, so we will just call it the mea. Also ote that the mea is usually deoted by a bar over the x. Here are a couple of examples. Example Fid the mea of the umbers,4,,0,4,5. The mea is x = =. Example Fid the mea of the umbers 1,,,1,,1,1345. The mea is x = = = 13. I Example 8.1.4, I thik you will agree that the mea does ot really give a useful measure of what the average of the umbers is. 1

2 It is to get over this sort of problem that we will defie two other sorts of average. While oe of them is perfect, if we give all three, we will get a reasoable idea of what the average is. The ext defiitio is as follows. Defiitio Media. Give a list of umbers x 1,x,...,x i ascedig order, the their media is defied to be m = x +x +1 if is eve ad m = x+1 if is odd. Remark This defiitio may seem a bit complicated but all it is sayig is that the media is the umber that is half way alog the list, where we have to allow for the fact that if there are a eve umber of umbers there is o oe umber half way alog the list, so we take the mea of the oes before ad after half way. For example if we have four umbers x 1,x,x 3,x 4 i ascedig order, the = 4 = x4 +x4 +1 so it is eve ad the media is defied to be x +x +1 as we wat. = x +x 3, O the other had if is odd, say = 5, the the media of the umbers x 1,x,x 3,x 4,x 5 which are i ascedig order is x+1 = x5+1 = x 3, agai as we wat. Here are some examples. Example Fid the media of the umbers 3,4,,0,4,5,,1,0,,8. Whe asked to fid the media of a list of umbers, the usually it is a good idea to first put them i ascedig order, sice this makes it much easier to see where the middle oe or middle two if there are a eve umber of umbers lies. This is best doe by crossig out the umbers as you write each ew umber dow. I this case the ew list is 3,,,0,0,1,,4,4,5,8. Sice there are 11 umbers a odd umber, the media is m = x11+1 = x = 1. Note that we do t have to write all the umbers dow; sice we oly wat x we oly eed to write the first six dow. If you are short of time the this may be a good idea but it ca also be useful to write them all dow to check you have ot missed ay. Example Fid the media of the umbers 10,8,,,4,5,, 1,0, 4,5,. Here there are twelve umbers a eve umber of umbers, so we are lookig for m = x1 +x1 +1 = x +x. The first seve umbers i ascedig order are, 4, 1,0,,,4. Thus the media is m = +4 = 3. Example Fid the media of the umbers 1,,,1,,1,1345. Here there are seve umbers a odd umber of umbers, so we are lookig for m = x+1 = x 4. The first four umbers i ascedig order are 1,1,1,. Thus the media is.

3 Remark The umbers i Examples ad are the same. I thik you will agree that it could be argued that the media gives a better measure of a represetative umber i this case tha the mea. There is oe other measure of average that we will look at i this sectio. Defiitio Mode. Give a list of umbers, the their mode is defied to be the umber or umbers that occur most frequetly. Remark Note that i cotrast to the mea ad media, the mode of a list of umbers may ot be uique. That is there may be more tha oe mode. The mode also makes sese for o-umeric data. For example, we could fid the mode of the types of car that pass us i the street i oe hour, or the mode of the first ames of the people i a classroom. We wo t do this i this course however. Here are some examples. Example Fid the mode of the umbers 1, 3,0,3,4,5,5,. Here we have two fives but oly oe of each of the other umbers, so the mode of this list is 5. Example Fid the mode of the umbers 1,1,,,3,4,5. Here we have two oes ad two twos but oly oe of each of the other umbers, so 1 ad are both modes of this list. Example Fid the mode of the umbers 1,0,1,,3,4,5. Here there is oe of each umber, so each of the give umbers is a mode of the list. 8.. Measures of Spread: Stadard Deviatio, Variace ad Iterquartile Rage. I Sectio 8.1 we looked at various measures of the average of a list of umbers; that is what umbers give us a good idea of what a typical umber is. Aother importat property of a list of umbers is how spread out they are ad it i this sectio we will look at some measures of this. Oe way of measurig the spread of a list of umbers would be to calculate the mea ad the calculate the mea of the distaces of the umbers from the mea. However this measure is ot usually used; istead the followig measure is much more commo. Defiitio 8..1 Stadard Deviatio. Give a list of umbers x 1,x,...,x, the their stadard deviatio is defied to be x σ =. 3

4 Remark 8... The stadard deviatio is usually deoted by the Greek letter sigma. Let us have a look at what this meas i words. We first fid the mea x, the calculate the sum of the squares of the differeces of the umbers from the mea, the divide by the umber of umbers ad fially take the square root. The is aother form of stadard deviatio the sample stadard deviatio where we divide by 1 rather tha. This is used whe calculatig the stadard deviatio of a sample rather tha the whole populatio i a opiio poll for example. We will ot calculate this i this course. The quatity x also has a ame. Defiitio 8..3 Variace. Give a list of umbers x 1,x,...,x, the their variace is defied to be Varx = σ = x. Here are a couple of examples of calculatig the variace ad stadard deviatio. Example Fid the variace ad stadard deviatio of the umbers,4,,0,4,5. Usig Example 8.1.3, the mea is x = 13. Hece the variace is Varx = = x 13 = Thus the stadard deviatio is σ = Varx = Example Fid the variace ad stadard deviatio of the umbers 1,,,1,,1,

5 Usig Example 8.1.4, the mea is x = Hece the variace is x Varx = = Thus the stadard deviatio is σ = Varx While the stadard deviatio i Example 8..5 does show that the umbers are more spread out tha the umbers i Example 8..4, it could be argued that this is all due to oe umber. If the umbers are data from some experimet, for example, it is probable that the umber 1345 is due to some sort of error ad i this case we really eed a better measure of spread. If we compare Example to Example 8.1.4, we see that i this case the media gives a better measure of cetre tha the mea. So it might be expected that a better measure of spread will be related to the media, rather tha the mea which the stadard deviatio is related to. I fact this will be the case but before we give the defiitio of this ew measure of spread, we eed the followig. Defiitio 8.. Lower ad Upper Quartile. Give a list of umbers, the their lower quartile ad upper quartile are calculated as follows: 1 List the umbers i ascedig order. If there are a eve umber of umbers, the split the umbers ito a lower half ad ad upper half. If there are a odd umber of umbers, the discard the media ad split the remaiig umbers ito a lower half ad a upper half. 3 The lower quartile, deoted Q 1, is the media of the lower half of umbers. The upper quartile, deoted Q 3, is the media of the upper half of umbers. Warig 8... Ufortuately there is o geerally accepted way to calculate the lower ad upper quartile sometimes also called the first ad third quartiles. So, if you are readig ay particular book, you first have to make sure exactly what methods are beig used to calculate them Remark Sometimes the media is called the secod quartile ad is deoted Q. We ca ow defie our ew measure of spread. 5

6 Defiitio 8..9 Iterquartile Rage. Give a list of umbers, the their iterquartile rage is defied to be Q 3 Q 1. Here are some examples. Example Fid the iterquartile rage of the umbers 1,,,1,,1,1345. We first write the umbers i ascedig order: 1,1,1,,,,1345. Sice there are seve umbers aoddumber themediaisgivebyx+1 = x 4 = otewedo t actually eed to kow what the media is, just where it lies i the list. Agai, sice we have a odd umber of umbers, we discard the media ad split the remaiig umbers ito a lower half 1,1,1 ad a upper half,,1345. There are three umbers i each of these ew groups a odd umber, so i each case the media is x3+1 = x. Thus the lower quartile is Q 1 = 1 ad the upper quartile is Q 3 =. Hece the iterquartile rage is Q 3 Q 1 = 1 = 1. Remark The umbers i Examples 8..5 ad are the same ad I thik you will agree that i some situatios, the iterquartile rage is a better measure of spread tha the stadard deviatio. Example Fid the iterquartile rage of the umbers,3, 1,,0,1,3,4. We first write the umbers i ascedig order:,, 1,0,1,3,3,4,. Sice there are ie umbers a odd umber the media is give by x9+1 = x 5 = 1. Agai, sice we have a odd umber of umbers, we discard the media ad split the remaiig umbers ito a lower half,, 1,0 ad a upper half 3,3,4,. This time there are four umbers i each of these ew groups a eve umber, so +x4 +1 = 3 adtheupperquartile isq 3 = 3+4 i each case the media is x4 + 1 rage is Q 3 Q 1 = 3 = 5. = x +x 3. Thus the lower quartile is Q 1 = =. Hece theiterquartile Example Fid the iterquartile rage of the umbers 9,8,3,5,, 4, 8, 3. We first write the umbers i ascedig order: 9, 8, 4, 3,,3,5,8. Sice there are eight umbers a eve umber we just split the umbers ito a lower half 9, 8, 4, 3 ad a upper half,3,5,8. Agai there are four umbers i each of these ew groups a eve umber, so the media is x4 +x4 +1 = x +x 3. Hece the lower quartile is Q 1 = 8+ 4 = ad the upper quartile is Q 3 = 3+5 = 4. Thus the iterquartile rage is Q 3 Q 1 = 4 = 10. Example Fid the iterquartile rage of the umbers 0,3,, 11,14,4,1, 1, 1,4. Wefirstwritetheumbers iascedig order: 11,, 1, 1,0,1,3,4,4,14. Sice

7 there are te umbers a eve umber we just split the umbers ito a lower half 11,, 1, 1,0 ad a upper half 1,3,4,4,14. This time there are five umbers i each of these ew groups a odd umber, so the media is x5+1 = x 3. Hece the lower quartile is Q 1 = 1 ad the upper quartile is Q 3 = 4. Thus the iterquartile rage is Q 3 Q 1 = 4 1 = Lie of Best Fit: Least Squares. So far i this chapter we have just looked at situatios where our data is a list of umbers. I this sectio we will look at the case where our data is a list of poits i the x-y plae. Give a list of poits like this, we might ask ourselves what is the best lie we ca draw to represet these poits. There are several ways we could calculate the lie of best fit. For example, if we have poits which we will deote by,y i, i = 1,,...,, the oe way to fid the best lie would be to fid a lie y = mx + c such that the quatity m + c y i is miimized. That is the lie such that the sum of the vertical distaces of the poits from the lie is miimized. However a much more commo method is to miimize the sum of the squares of these distaces. Note that this is somewhat similar to the calculatio of the variace, where we fid the sum of the squares of the distaces from the mea. The actual derivatio of the formulae that allow us to calculate m ad c usig this method is quite complicated, so I will just state them. Theorem Lie of best fit: least squares. Give a list of poits,y i, i = 1,,...,, the the lie of best fit y = mx+c calculated usig the least squares method is foud usig the formulae: m = y i x i Here are a couple of examples. y i ad c = y mx. Example Fid the lie of best fit usig the least squares method with the poits 3, 1,,0,1,1,3,,4,3,,5,8,,11,,1,5 ad 14,5. Plot the lie of best fit ad the poits o a graph. I this case = 10 ad 10 = = = y i = y i = =

8 10 y i = y i 10 x i = = = =08. x i = = =00. Hece y i m = ad x i y i = 10 y i = = c = y mx = 10 m 10 = = Thus the lie of best fit is y = 3 14 x The poits ad the graph are show i Figure 1. Figure 1. The Lie of Best Fit ad Poits From Example

9 Remark As you ca see i Figure 1, the lie does fit the poits quite well. Example Fid the lie of best fit usig the least squares method with the poits 3,4, 1,4, 1,3,1,,,3,3,1,,0,8,1,9, 1,11, 1 ad 13,. Plot the lie of best fit ad the poits o a graph. I this case = 11 ad 11 = = = y i = y i = = y i = y i = = = x i = x i = = =49. Hece m = y i x i = = = , 9 y i

10 ad c = y mx = 11 y i 11 m = Thus the lie of best fit is y = x The poits ad the graph are show i Figure = Figure. The Lie of Best Fit ad Poits From Example

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Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the