# Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

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1 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 we will go o to the case of a frequecy distributio. The first thig to ote is that, whereas the rage as well as the quartile deviatio are two such measures of dispersio which are NOT based o all the values, the mea deviatio ad the stadard deviatio are two such measures of dispersio that ivolve each ad every data-value i their computatio. You must have oted that the rage was measurig the dispersio of the data-set aroud the mid-rage, whereas the quartile deviatio was measurig the dispersio of the data-set aroud the media. How are we to decide upo the amout of dispersio roud the arithmetic mea? It would seem reasoable to compute the DISTANCE of each observed value i the series from the arithmetic mea of the series. Let us do this for a simple data-set show below: The Number of Fatalities i Motorway Accidets i oe Week: Day Number of fatalities Suday 4 Moday 6 Tuesday Wedesday 0 Thursday 3 Friday 5 Saturday 8 Total 8 Let us do this for a simple data-set show below: The Number of Fatalities i Motorway Accidets i oe Week: Day Number of fatalities Suday 4 Moday 6 Tuesday Wedesday 0 Thursday 3 Friday 5 Saturday 8 Total 8

2 The arithmetic mea umber of fatalities per day is I order to determie the distaces of the data-values from the mea, we subtract our value of the arithmetic mea from each daily figure, ad this gives us the deviatios that occur i the third colum of the table below: Day Number of fatalities Suday 4 0 Moday 6 + Tuesday Wedesday 0 4 Thursday 3 1 Friday Saturday TOTAL 8 0 The deviatios are egative whe the daily figure is less tha the mea (4 accidets) ad positive whe the figure is higher tha the mea. It does seem, however, that our efforts for computig the dispersio of this data set have bee i vai, for we fid that the total amout of dispersio obtaied by summig the (x x) colum comes out to be zero! I fact, this should be o surprise, for it is a basic property of the arithmetic mea that:the sum of the deviatios of the values from the mea is zero. The questio arises: How will we measure the dispersio that is actually preset i our data-set? Our problem might at first sight seem irresolvable, for by this criterio it appears that o series has ay dispersio. Yet we kow that this is absolutely icorrect, ad we must thik of some other way of hadlig this situatio. Surely, we might look at the umerical differece betwee the mea ad the daily fatality figures without cosiderig whether these are positive or egative. Let us deote these absolute differeces by modulus of d or mod d. This is evidet from the third colum of the table below: d d Total 14

3 By igorig the sig of the deviatios we have achieved a o-zero sum i our secod colum. Averagig these absolute differeces, we obtai a measure of dispersio kow as the mea deviatio. I other words, the mea deviatio is give by the formula: MEAN DEVIATION: M.D. d i As we are averagig the absolute deviatios of the observatios from their mea, therefore the complete ame of this measure is mea absolute deviatio --- but geerally we just say mea deviatio. Applyig this formula i our example, we fid that: The mea deviatio of the umber of fatalities is 14 M.D.. 7 The formula that we have just cosidered is valid i the case of raw data. I case of grouped data i.e. a frequecy distributio, the formula becomes MEAN DEVIATION FOR GROUPED DATA: M.D. f i x i x fi d i As far as the graphical represetatio of the mea deviatio is cocered, it ca be depicted by a horizotal lie segmet draw below the -axis o the graph of the frequecy distributio, as show below:

5 Variace ( x x) Let us compute this quatity for the data of the above example. Our -values were: Takig the deviatios of the -values from their mea, ad the squarig these deviatios, we obtai: ( x x ) ( x x ) Obviously, both ( ) ad () equal 4, both ( 4) ad (4) equal 16, ad both ( 1) ad (1) 1.

6 Hece (x x) 4 is ow positive, ad this positive value has bee achieved without bedig the rules of mathematics. Averagig these squared deviatios, the variace is give by: Variace: ( x x) The variace is frequetly employed i statistical work, but it should be oted that the figure achieved is i squared uits of measuremet. I the example that we have just cosidered, the variace has come out to be 6 squared fatalities, which does ot seem to make much sese! I order to obtai a aswer which is i the origial uit of measuremet, we take the positive square root of the variace. The result is kow as the stadard deviatio. STANDARD DEVIATION: S ( x x ) Hece, i this example, our stadard deviatio has come out to be.45 fatalities. I computig the stadard deviatio (or variace) it ca be tedious to first ascertai the arithmetic mea of a series, the subtract it from each value of the variable i the series, ad fially to square each deviatio ad the sum. It is very much more straight-forward to use the short cut formula give below: SHORT CUT FORMULA FOR THE STANDARD DEVIATION: S x x I order to apply the short cut formula, we require oly the aggregate of the series ( x) ad the aggregate of the squares of the idividual values i the series ( x). I other words, oly two colums of figures are called for. The umber of idividual calculatios is also cosiderably reduced, as see below:

7 Total Therefore S ( 16) 6.45 fatalities The formulae that we have just discussed are valid i case of raw data. I case of grouped data i.e. a frequecy distributio, each squared deviatio roud the mea must be multiplied by the appropriate frequecy figure i.e. STANDARD DEVIATION IN CASE OF GROUPED DATA: S f ( x x ) Ad the short cut formula i case of a frequecy distributio is: SHORT CUT FORMULA OF THE STANDARD DEVIATION IN CASE OF GROUPED DATA: fx fx S Which is agai preferred from the computatioal stadpoit? For example, the stadard deviatio life of a batch of electric light bulbs would be calculated as follows: EAMPLE: Life (i Hudreds of Hours) No. of Bulbs f Midpoit x fx fx ad over

8 Therefore, stadard deviatio: S hudredhours 1390 hours As far as the graphical represetatio of the stadard deviatio is cocered, a horizotal lie segmet is draw below the -axis o the graph of the frequecy distributio --- just as i the case of the mea deviatio. f Stadard deviatio The stadard deviatio is a absolute measure of dispersio. Its relative measure called coefficiet of stadard deviatio is defied as: Coefficiet of S.D: Sta dard Deviatio Mea

10 Utreated lad: per cet 35 Treated lad: per cet 58 The coefficiet of variatio for the utreated lad has come out to be 8.57 percet, whereas the coefficiet of variatio for the treated lad is oly 17.4 percet.

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