Descriptive statistics deals with the description or simple analysis of population or sample data.

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1 Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small part of it or sample is examied istead. If a sample is represetative of the populatio, a lot ca be leared about the populatio by studyig the sample. Descriptive statistics deals with the descriptio or simple aalysis of populatio or sample data. Perhaps the simplest aalysis of data is to classify idividual observatios ito two or more categories accordig to some attribute that they possess, e.g. people ca be classified ito employed/uemployed, or ito several groups accordig to religio, etc. A variable is a measurable characteristic which chages from oe member of a sample or a populatio to aother, e.g. age of a perso, umber of defective items i a stadard box of screws, GDP of a coutry, etc. A cotiuous variable is a measurable characteristic which potetially ca take ay value i a cotiuous rage, without ay breaks or jumps. Height ad icome of a perso are examples of cotiuous variables. A discrete variable is a measurable characteristic which is restricted to a specific set of values. (Which ca be fiite or ifiite). Number of people i a family, umber of defective lightbulbs i a load, score o a die, etc. are examples of discrete variables because, i these cases, oly iteger values are possible. Note that a variable would still be discrete if it could potetially take ay iteger value, however high the importat thig beig there are discrete jumps i the value. The discrete jumps do t have to be iteger, either; a variable that could take the values (say) 0, 0.25, 0.5, 0.75 or ad oe other would still be discrete. Whe observatios are classified ito a umber of class itervals specified i terms of a variable, we have a frequecy distributio showig e.g. that there are 2 people i a class with height below 5, 4 people with height

2 betwee 5 ad 5 2, 7 people with height betwee 5 2 ad 5 4, etc. Such distributios ca be represeted i the form of a table or diagram. Discrete data are ofte represeted by bar charts, showig e.g. umber of families with 0,,2, childre. Cotiuous data are usually represeted as histograms, showig frequecy desity per uit of the variable so for example if we had 8 people with icomes betwee 0,000 ad 4,000, ad 4 people with icomes betwee 4,000 ad 6,000, these two categories would have the same frequecy desity (twice as may people, but twice as wide a category), ad therefore the same height i a histogram. The first category though would have twice the width i the histogram. It is importat to distiguish betwee exact limits of a class iterval ad groupig limits, e.g. icomes are ofte recorder to the earest poud, ad for groupig purposes a class iterval may be specified as cotaiig (say) icome betwee 50 ad 99 p.w. (icludig both values). The exact limits for the iterval are ad p.w. because of roudig. Exact limits are importat for aalytic ad graphical purposes. Measures of Cetral Tedecy For comparative purposes, it is ofte desirable to represet (summarise) a frequecy distributio by a sigle value. Mea, media ad mode are the most commoly used measures of cetral tedecy. The mea (arithmetic mea) or average is defied as: = ( i ) / Where i,,, are idividual observatios, there beig of them. ( is called the sample size). For grouped data we have = r r f i f i i

3 Where i are midpoits of exact class itervals (there beig r of them) ad f i are class frequecies. The mea is easy to uderstad ad has a umber of properties which are importat for aalytical ad empirical work. For example, if is the average icome of people, their total icome is give by *. So ca be see as the icome everyoe would have if all the icome i the group were distributed equally. Also, ( i ) =0, i.e. deviatios aroud the mea sum to zero. A disadvatage of the mea is that it is greatly iflueced by extreme observatios. The media is defied as the middle value whe observatios are arraged i a ascedig or descedig order. Whe the umber of observatios is eve, it is the average of the two middle values. For grouped data (frequecy table), the media is give by Me = m L m + cm fi / 2 f m Where L m = exact lower limit of class iterval, c m = width of the media class iterval, (/2) = rak order of the media, f i =cumulative frequecy up to but ot icludig media category, f m = frequecy of the media class iterval. The media is easy to uderstad, ad it is ot greatly affected by extreme observatios. It is ofte used i preferece to the mea whe extreme observatios are preset or whe a distributio is heavily skewed, e.g. distributio of icome or wealth. The mode is defied as the most commo value of idividually recorder observatios ad as the value of the variable for which the frequecy desity is greatest for grouped data. Media ad Mea m

4 I skewed distributios, these two statistics will differ. For example, i the case of icomes, a few people with very high icomes push up the mea, but do t affect the media, so that the mea is higher. I other words, most people have below average icomes! O the other had if we cosider average life expectacy, a small umber of people who die i ifacy (compared to very, very few ideed who live to, say, 40) push the mea dow. (This will especially be true i developig coutries). So, most people live loger tha the average life expectacy. For symmetric distributios, mea, media ad mode all coicide, assumig middle values are more likely tha extreme values. For skewed distributios, we ca see the relatioship betwee the three measures graphically: Fr Positively skewed distributio Negatively skewed distributio Fr Mode Media Mea Mea Media Mode Measures of variability 2.3 Variability of measuremets is ofte more importat i statistical aalysis tha cetral tedecy. Three measures of variability are discussed below. Mea deviatio is defied as

5 MD = i / (For ugrouped data), where the. sig meas absolute value, that is igorig mius sigs. This measure, while ituitively atural, is ot much use for aalytical work because of the modulus sigs it is ot a smooth measure. (see below) Stadard deviatio is defied as: S = 2 ( i ) (for ugrouped data) S 2 is called the variace. Stadard deviatio avoids the modulus operator, ad is measured i the same uits as the data itself, ad as the mea. It ca be iterpreted as a typical deviatio from the mea. The variace is the average squared deviatio from the mea. The stadard deviatio is its square root. Example Te studets i a class have test scores of 35, 42, 46, 5, 54, 59, 64, 66, 73 ad 78.

6 The mea score is ( )/0=56.8. The deviatios from the mea are therefore -2.8,-4.8,-2.8,-5.8,- 2.8,+2.2,+7.2,+9.2,+6.2,+2.2. To fid the variace, we square each of these values (to get ,29.04, ), add them up ad divide by 0, to get a average of Fially to fid the stadard deviatio, we take the square root of the variace, givig a value of Coefficiet of variatio, defied as S/, is a relative measure of dispersio used to compare variability of two or more distributios whose meas ad stadard deviatios differ a great deal; for example, the stadard deviatio of icomes, measured i GB, i the UK would be a lot higher tha i, say, a poor Africa coutry. However, if we scale this by the mea to get the CV, this could be greater or smaller i either coutry. Skewess May distributios are ot symmetrical ad their degree of skewess ca vary quite cosiderably, e.g. the distributio of wealth is more skewed tha the distributio of icome. Asymmetry ca be measured i a umber of ways. Oly oe measure will be metioed here. The Pearso coefficiet of skewess is defied as SK = ( Mo) / S 3( Me) / S The differece -Mo, which is approximately equal to -Me i a ot-tooskewed distributio, icreases with skewess. Divisio by S esures that it is ot affected by chages i uits of measuremet ad variability of differet distributios. Note that for positively skewed distributios, - Mo is positive, ad for a egatively skewed distributio it is egative. The absolute value of SK ca be as large as 3, but i practice values larger tha are rare.