Introduction to descriptive statistics

1 Introduction toDescriptive StatisticsJackie NicholasMathematics Learning CentreUniversity of SydneyNSW 2006c 1999 University of SydneyAcknowledgementsParts of this booklet were previously published in a booklet of the same name by theMathematics Learning Centre in 1990. The rest is wish to thank the Sue Gordon for her numerous suggestions about the content and bothSue and Usha Sridhar for their careful NicholasJanuary 1999 Contents1 Measures of Central The Mean, Median and 32 Measures of The Range .. Standard Deviation .. The Interquartile Range .. Quartiles for small data sets .. The interquartile 123 Formulae for the Mean and Standard Formulae for Mean and Standard Deviation of a Estimates of the Mean and Variance.

2 154 Presenting Data Using Histograms and Bar .. Histograms .. Constructing Histograms and Bar Graphs from Raw Data .. 255 The Constructing a Box-plot .. Using Box-plots to Compare Data 306 Solutions to Solutions to Exercises from Chapter 1 .. Solutions to Exercises from Chapter 2 .. Solutions to Exercises from Chapter 3 .. Solutions to Exercises from Chapter 4 .. Solutions to Exercises from Chapter 5 .. 36 mathematics Learning Centre, University of Sydney11 Measures of Central The Mean, Median and ModeWhen given a set of raw data one of the most useful ways of summarising that data is tofind an average of that set of data. An average is a measure of the centre of the data are three common ways of describing the centre of a set of numbers.

3 They are themean, the median and the mode and are calculated as mean add up all the numbers and divide by how many numbers there median is the middle number. It is found by putting the numbers in orderand taking the actual middle number if there is one,or the average of the two middle numbers if mode is the most commonly occurring s illustrate these by calculating the mean, median and mode for the following of luggage presented by airline passengers at the check-in (measured to the nearestkg).1823202124232020151924 Mean =18+23+20+21+24+23+20+20+15+19+2411= = middle valueMode = number 20 occurs here 3 the mean, median and mode are all appropriate measures of central tendency describes the tendency of the observations to bunch around a particularvalue, or category.

4 The mean, median and mode are all measures of central are all measures of the average of the distribution. The best one to use in a givensituation depends on the type of variable Learning Centre, University of Sydney2 For example, suppose a class of 20 students own among them a total of 17 pets as shownin the following table. Which measure of central tendency should we use here?Type of PetNumberCat5 Dog4 Goldfish3 Rabbit1 Bird4If our focus of interest were on thetypeof pet owned, we would use the mode as ouraverage. Cat would be described as the modal category , as this is the category thatoccurs most , on the other hand, we were not interested in the type of pet kept but the averagenumberof pets owned then the mean would be an appropriate measure of central tendency.

5 Herethe mean is1720= , if we are interested in the average number of pets per student then our data mightbe presented quite differently as in the table of PetsTallyFrequency0|||| |||| |111||||42|||33|14|1 Now we are concerned only with a quantity variable and the average used most commonlywith quantity variables is the mean. Here, again, the mean is =(11 0) + (4 1)+(3 2)+(1 3) + (1 4)20= that (4 1) is really1+1+1+1,since 4 students have 1 pet each, and (3 2) isreally 2 + 2 + 2, since 3 students have 2 pets each. Since there are 20 scores the medianscore will occur between the tenth and the eleventh score. The median is 0, since thetenth and the eleventh scores are both 0, and the mode is mean has some advantages over the median as a measure of central tendency ofquantity variables.

6 One of them is that all the observed values are used to calculate themean. However, to calculate the median, while all the observed values are used in theranking, only the middle or middle two values are used in the calculation. Another isthat the mean is fairly stable from sample to sample. This means that if we take severalsamples from the same population their means are less likely to vary than their Learning Centre, University of Sydney3 However, the median is used as a measure of central tendency if there are a few extremevalues observed. The mean is very sensitive to extreme values and it may not be anappropriate measure of central tendency in these cases. This is illustrated in the s look again at our pets example and suppose that one of the students kept 18 of PetsTallyFrequency0|||| |||| |111||||42||23|14|118|1 The mean is now , but the median and the mode are still 0.

7 The effect of the outlierwas to significantly increase the mean and now the median is a more accurate measure ofthe centre of the the exception of cases where there are obvious extreme values, the mean is the valueusually used to indicate the centre of a distribution. We can also think of the mean asthe balance point of a example, consider the distribution of students marks on a test given in Figure doing any calculation, we would guess the balance point of the distribution tobe approximately 58. (Think of it as the centre of a see-saw.)Figure 1: Students marks on a patients at a doctor s surgery wait for the following lengths of times to see mins17 mins8 mins2 mins55 mins9 mins22 mins11mins16 mins5 minsWhat are the mean, median and mode for these data? What measure of centraltendency would you use here?

8 7060504080 mathematics Learning Centre, University of is the appropriate measure of central tendency to use with these data?Method of TransportNumber of measure of central tendency is best used to measure the average house pricein Sydney? doing any calculation, estimate the mean of the distribution in Figure 2: Students marks on a Learning Centre, University of Sydney52 Measures of DispersionThe mean is the value usually used to indicate the centre of a distribution. If we aredealing with quantity variables our description of the data will not be complete withouta measure of the extent to which the observed values are spread out from the will consider several measures of dispersion and discuss the merits and pitfalls of The RangeOne very simple measure of dispersion is the range.

9 Lets consider the two distributionsgiven in Figures 3 and 4. They represent the marks of a group of thirty students on 3: Marks on test 4: Marks on test it is clear that the marks on test A are more spread out than the marks on test B,and we need a measure of dispersion that will accurately indicate test A, the range of marks is 70 45 = test B, the range of marks is 65 45 = the range gives us an accurate picture of the dispersion of the two , as a measure of dispersion the range is severely limited. Since it depends only ontwo observations, the lowest and the highest, we will get a misleading idea of dispersion ifthese values are outliers. This is illustrated very well if the students marks are distributedas in Figures 5 and 5: Marks on test Learning Centre, University of Sydney6 Figure 6: Marks on test test A, the range is still 70 45 = test B, the range is now 72 40 = 32, but apart from the outliers, the distribution ofmarks on test B is clearly less spread out than that of want a measure of dispersion that will accurately give a measure of the variability ofthe observations.

10 We will concentrate now on the measure of dispersion most commonlyused, the standard Standard DeviationSuppose we have a set of data where there is no variability in the observed values. Eachobservation would have the same value, say 3, 3, 3, 3 and the mean would be that samevalue, 3. Each observation would not be different ordeviatefrom the suppose we have a set of observations where there is variability. The observed valueswould deviate from the mean by varying standard deviationis a kind of average of these deviations from the is best explained by considering the following , for example, the following grades of 6 = find how much our observed values deviate from the mean, we subtract the mean values56 48 63 60 51 52 Deviations from Mean +1 7+8+5 4 3We cannot, at this stage, simply take the average of the deviations as their sum is zero.