4. Variation and standard deviation.pptx

8/20/2019 4. Variation and standard deviation.pptx

1/13

Variation and spread of

distribution


2/13

The normal distribution• A normal distribution is a distribution of scores that is peaked

in the middle and tails o symmetrically on either side of thepeak. The distribution is often said to be ‘bell-shaped’. For aperfectly normal distribution, the mean, median and mode willbe represented by the peak of the curve.


3/13

The normal distribution

• n everyday life, many variables such ashei!ht, wei!ht, shoe si"e, an#iety levels ande#am marks all tend to be normally

distributed.• For a distribution to be classed as normal it

should have the followin! characteristics$

t should be symmetrical about the mean.

The tails should meet the x-axis atinnity.

t should be bell-shaped.


4/13

The normal distribution

• All of thosedistributions arenormal, even thou!hthey are not the

same, they havecharacteristics$symmetrical, in%nityand bell-shaped.

• The only di&erence isin standard deviation,actually in howspread is the curve.


5/13

Variation or spread ofdistributions

• Another important aspect of a sampleor population of scores is how spreadout they are. 'r, to put it another way,

how much variation there is in yoursample or population.

• Variance or variation of scores

indicates the degree to which thescores on a variable are dierentfrom one another.


6/13

The ran!e

• 'ne simple way of !ettin! an indication ofthe spread of scores is to compare theminimum score with the ma#imum score in

the sample or population. This is known asthe range. The ran!e is simply the di&erencebetween the minimum and ma#imum scores.

• Althou!h the ran!e tells us about the overall

ran!e of scores, it does not !ive us anyindication of what is happenin! in betweenthese scores.


7/13

•

These histograms were generated from two sets of data which havethe same mean (!" and the same minimum and maximum scores(# and $%".

• They both therefore have the same range, which is $$ ($% minus #".They are, however, totally dierent distributions& the scores indistribution ' are packed tightly around the mean whereas the

scores in distribution are generally more spread out.


8/13

(tandard deviation•

A more informative measure of the variation in data isthe standard deviation ()*". +ne of the problems withthe ran!e is that it does not tell us what is happenin!with the scores between the minimum and ma#imumscores. The (), however, does !ive us an indication of

what is happenin! between the two e#tremes. Thereason why the () is able to do this is that it tells ushow much all the scores in a dataset vary around themean. t is important because it forms the basis ofmany of the statistical techni*ues we use to analyseour data.

• The standard deviation is the degree to whichthe scores in a dataset deviate around themean. It is an estimate of the average deviation

of the scores from the mean.


9/13

+ow to calculate () • (uppose that we have the followin! !roup of scores

collected from a study into the number of chocolate barseaten by people each week$ /, 0, 1, 2, 3, //.

• To work out the standard deviation, we proceed as follows$

/. First, calculate the mean, which is 2.

4. The deviation of each score from the mean is$ 51, 54,5/, 6, 7, 1 if we add these up, you see that we !et "ero.

7. 8e therefore need to s*uare these deviations to !et rid ofthe ne!ative values, which !ives us these scores$ 41, 0,/, 6, 3, 41.

0. 9e#t, we calculate the mean of these scores, which is/6.2:, i.e. 20 ; 2, which !ives us our variance.

1. Finally, we work out the standard deviation by takin! thes*uare root of the variance, which !ives us 7.4:.


10/13

'ther characteristics of distributions• The kurtosis of a distribution is a measure of how peaked

the distribution is. at distribution is called platykurtic, avery peaked distribution is called leptokurtic, and adistribution between these e#tremes is called mesokurtic.

• n (


11/13

9on-normal distributions• )kewed distributions are those where the peak is shifted

away from the centre of the distribution and there is ane#tended tail on one of the sides of the peak. A negatively

skewed distribution is one where the peak has been shiftedto the ri!ht towards the hi!h numbers on the scale and the

tail is pointin! to the low number or even pointin! to the

negative numbers". positively skewed distribution has thepeak shifted left, towards the low numbers, and has the

tailed e#tended towards the hi!h numbers.

• A positive value su!!ests a positively skewed distribution,whereas a ne!ative value su!!ests a ne!atively skewed

distribution. A value of "ero tells you that your distributionis not skewed in either direction. Values of skewness around

about / or 5/ su!!est deviations from normality whichare too e#treme for us to use many of the statistical

techni*ues


12/13


13/13

9on-normal distributions

• A bimodaldistribution is onethat has two

pronounced peaks.t is suggestive ofthere being twodistinctpopulationsunderlyin! thedata.

Documents

4. Variation and standard deviation.pptx