Section 9: Presenting and describing quantitative data

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1 Section 9: Presenting and describing quantitative data Australian Catholic University 2014 ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced or used in any form or by any means graphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, information storage and retrieval systems without the written permission of the publishers. Disclaimer: No person should rely on the contents of this publication without first obtaining advice from a qualified professional person. This publication is distributed on the terms and understanding that: (1) the authors, consultants and editors are not responsible for the results of any actions taken on the basis of information in this publication, nor for any error in or omission from this publication; and (2) the publisher is not engaged in legal, accounting, professional or other advice or services. The publisher, and the authors, consultants and editors, expressly disclaim all and any liability and responsibility to any person, whether a purchaser or a reader of this publication or not, in respect of anything, and of the consequences of anything, done or omitted to be done by any such person in reliance, whether wholly or partially, upon the whole or any part of the contents of this publication. Without limiting the generality of the above, no author, consultant or editor shall have any responsibility for any act or omission of any other author, consultant or editor. MGMT617 Research Methods Section 9 1

2 Note. The methods considered here can be implemented using the spreadsheet Excel. Other common computer packages, such as SPSS, are easily available, as are good introductions, such as that of Coakes & Ong (2011), but Excel is so commonly available that it is the simplest choice. If you do decide to use Excel, you may need to add the Excel Data Analysis package into your computer. To do this, open an Excel worksheet, click the Office (or File) button, click Options at the bottom, select Add-ins, and tick the Analysis Toolpack box. 9.1 Data preparation Prescribed reading 9.9 Zikmund, WG Babin, BJ Carr, JC & Griffin, M 2013, Business research methods, 9 th edn, South- Western, Cengage Learning, Mason OH. Chapter 10, pp Unless the data set is extremely small, the first step in quantitative analysis is to get the data entered into a computer file for use with a given program. This means that the data must be consistently coded into a form suitable for the program. If there are categorical background variables, they can be coded numerically too, to avoid problems with file formatting, though this may not be essential. Unless the data set is extremely small, the first step in quantitative analysis is to get the data entered into a computer file for use with a given program. This means that the data must be consistently coded into a form suitable for the program. If there are categorical background variables, they can be coded numerically too, to avoid problems with file formatting, though this may not be essential. Bhattacherjee (2012, p. 119) recommends producing a code book at this stage. The code book is a complete and detailed description of the variables in the study, with a record of how they were coded. For example, responses on a 5-point Likert scale, with possible responses Strongly disagree, Disagree, Undecided, Agree and Strongly Agree can be coded 1 to 5. If alternating positive and negative wording of items has been introduced to mitigate acquiescence bias, then some scoring will have to be reversed. If it is feasible, independent checking at different times or by different people should be done (Fink, 2003, p. 9). Zikmund et al. (pp ) discuss decisions about missing data and inconsistencies. Missing values in any variable require a decision. First, one must decide on a distinctive code for them, and then on how to treat the observation. Essentially, the choice is how far to exclude the data from an observation with one entry missing. Removing the whole observation can restrict the sample unduly, so the entry is sometimes replaced by the mean of that item. All decisions need to be carefully recorded. MGMT617 Research Methods Section 9 2

3 Frequency 9.2 Data presentation Prescribed reading 9.2 Zikmund, WG Babin, BJ Carr, JC & Griffin, M 2013, Business research methods, 9 th edn, South- Western, Cengage Learning, Mason OH. Chapter 19. Categorical data Frequency tables and bar graphs For categorical data one counts the number of cases in each category. For simplicity, assume the categories are Partnership, Proprietary Company, and Public Company, and that they have been entered case by case in Column A, cells 1 to 50, of an Excel spreadsheet. To get a frequency table, first list the categories in cells B1 to B3. In cell C1, use COUNTIF(A1:A50, Partnership ), and work similarly for the other two categories. Suppose the frequency table is as follows Category Frequency Partnership 16 Proprietary Company 22 Public Company 12 To display the information, highlight the two columns, go to the Insert tab and choose a column graph. Use the layout button for access to labels. Frequencies Partnership Proprietary Company Public Company Type Comparisons between groups can be represented by double column bar graphs. MGMT617 Research Methods Section 9 3

4 One may wish to use relative frequencies, that is, the proportion in each category. An alternative display in these terms is a pie chart, where proportions are represented as sectors of a circle of corresponding area. Contingency tables Relationships between categorical variables are usually presented as a contingency table. For example, suppose data for staff in a given organisation includes gender and citizenship. These categories can be represented in the following table. Australian Other Female Male Quantitative data Zikmund et al. (ch. 17) review the terminology and calculation methods needed here. We need a few more graphical techniques and the calculation of descriptive statistics. Frequency distributions and histograms Frequencies for quantitative scores are mostly presented in a histogram. A histogram is like a bar graph, but without gaps between columns. But if scores cover a wide range, with low frequencies at each point, you may wish to block your data into classes, because, without this, it is difficult to get an overall idea of the data. Look at the data range and decide how many non-overlapping classes you need. For example, exam marks between 0 and 100 could be blocked in 10s. Once you have done this, the Excel Data Analysis package will form a frequency table and produce a histogram for you. It will do the whole job for you, as in the example immediately below. Example Data set: Raw marks for 40 students in a statistics examination MGMT617 Research Methods Section 9 4

5 Relative frequency In this case, values range from 30 to 93, and frequencies for individual marks would be low. So here one would group them for display, as follows. Choose a number n of classes, and split the range into n intervals of equal size. The scores range from 32 to 91, with the difference being 59. But they actually include 60 values. So suppose we choose 5 non-overlapping classes each containing 12 possible values, with boundaries and frequencies as listed. It is not necessary to have the classes of equal width, but it is usually done this way to make interpretation easier. So, first put the raw data into, say, cells A3 to A42. Put the top score from each class interval into cells B3 to B7. Click on Data Analysis and select Histogram. The dialog box that comes up asks for the data range, and you select A3 to A42. Then you move to Bin Range, and select B3 to B6. Then you move to Output Range and select a position to start the output in. Then tick the box Chart Output, and click OK. The frequencies and the chart appear, and you remove gaps between columns in the chart by rightclicking on a column, selecting format Data Series, and choosing No Gap. Frequencies for the grouped data, and the histogram, appear below. Bins Frequency Histogram: Marks, grouped data Series Mark interval MGMT617 Research Methods Section 9 5

6 Frequency Frequency Histogram shapes Many of the methods of analysis that are used with quantitative data depend on whether the data set is approximately symmetrical. The histogram above is approximately symmetrical. A histogram with a long tail is called skewed. Examples are given below. Right skew ed distribution Score Series1 Left skew ed distribution Score Series1 Scatter plot Scatter plots are useful for checking whether there is a linear relationship between two variables measured for the same sample. Each pair of values is plotted as a pair of coordinates for a point in the plane. The resulting set of points can indicate relationships, some of which will be studied later. Example Data set: Examination marks for 15 students in Accounting and Business law Accounting Law MGMT617 Research Methods Section 9 6

7 Law Scatter plot Scatterplot: marks Accounting Series1 The diagram indicates that there seems to be an approximately linear relationship with higher scores on both variables going together. In Excel, put the pairs in two columns and highlight them. Click Insert, and select Scatter. 9.3 Descriptive statistics Prescribed reading 9.3 Zikmund, WG Babin, BJ Carr, JC & Griffin, M 2013, Business research methods, 9 th edn, South- Western, Cengage Learning, Mason OH. Chapter 17 & 20. Here we give calculations for summary statistics for quantitative data sets. Rank order calculations Quantitative data are ordered, so that one can obtain numerical measurements from the distribution of scores that give some idea of the location and variability of the scores. We have mentioned different types of quantitative data, and Zikmund et al. (pp ) go into some detail about this. For our purposes, it is enough to be aware that numbers representing rank order only should be described in terms of rank order only. Hence the rank order calculations described below can be used with all quantitative data, but the distance related calculations that follow are not applicable if the data represent only ranks. MGMT617 Research Methods Section 9 7

8 Percentiles, quartiles, median Suppose one has observations x 1, x 2,, x n. Then the p-th percentile of the distribution is the number c p such that p percent of the scores x 1 x n are less than or equal to c p, and 100-p percent are greater than or equal to c p. For our purposes the emphasis will be on the percentiles that split the distribution into quarters, with terminology as follows. The first quartile of a distribution is its 25 th percentile The median (or second quartile) of a distribution is its 50 th percentile. The third quartile of a distribution is its 75 th percentile. Excel contains functions MEDIAN and QUARTILE. The five-number summary for the data is the set Minimum, First quartile, Median, Third quartile, Maximum The median is used as a measure of central tendency. Distance-related calculations The range of a data set is the difference between the maximum and minimum scores. We now examine one aspect of the variability of a data set that is defined with reference to the quartiles. The inter-quartile range is the difference between the first and third quartiles. Note that it involves distance, not just rank, and is an indicator of general variability in the data set. An outlier in a data set is a score that is very different from the rest. The inter-quartile range is often used as a guide to identifying outliers. The most common test is done as follows. Let q 1 be the first quartile, and q 3 the third, and let r be the inter-quartile range. Make an interval bounded below by q r, and above by q r. Any member of the data set outside this interval qualifies as an outlier. Different software packages may use different formulae for the boundaries of the interval, but for our purposes, the rule above is quite sufficient. Distance-related calculations Mean and standard deviation Note. The symbol x i MGMT617 Research Methods Section 9 8

9 means the sum of the values x 1,, x n. The mean of a data set is just the average of the scores. So if the scores are x 1,, x n, then the mean is the number = (x 1 + x x n )/n = (1/n) x i In Excel, use AVERAGE Both the mean and the median give ideas of the central location of the distribution. Whereas the median depends only on rank order, the mean gives weight to the size of the scores. If the distribution is exactly symmetrical, they are the same. If the scores are strongly skewed, the median may give a better idea of what is typical. Most economic data, such as incomes, are skewed, because the typical person is poor compared with the few millionaires, the size of whose income increases the mean, but has no effect on the median. The variance of a set of scores x 1,, x n is the number s 2 = (x i - x) 2 /(n-1) and the standard deviation s is the square root of the variance. In Excel, use STDEV The variance indicates the variability of the data set. Its formulation in squares means that variation in different directions do not cancel out. So, if the histogram is approximately symmetrical, use the mean and standard deviation as summary numbers, but if the histogram is markedly skewed, use the median and inter-quartile range. 9.4 The normal distribution Many physical measurements follow a normal probability distribution. In this case, the proportion of the population with scores between numbers A and B is equals the area between A and B under the curve with equation f(x) = {1/[ (2 )]} exp [ -(x- ) 2 /(2 2 )] where exp(x) = e x. The formula looks terrible, and is terrible, but it is extremely important. The graph of f(x) is a symmetrical bell shaped curve, centred on the value, as sketched in Figure 5.2 with = 3 and = 2. These are the mean and the standard deviation for the population. There is a normal distribution for every possible pair of values and, which are the parameters of the distribution. MGMT617 Research Methods Section 9 9

10 Density function Normal curve x The probability of a score of at most A is the area under the curve as far as the x-value A. The difficulty of the formula means that probabilities have to be tabulated, and this is done for the special case where = 0 and = 1 (the standard normal distribution). You will not need to learn to use the table, because the Excel function NORMDIST calculates the probabilities. Most of the analyses that are commonly used assume that the variables being measured for the sample have a distribution close to normal on the whole population. It follows that sample data need to be examined for approximate normality before analyses are carried out. One can use the histogram as a rough guide, but it is a good idea also to produce a normal quantile plot for the data. Instructions for making a normal quantile plot in Excel are in the appendix to the next section. 9.5 References Bhattacherjee, A 2012, Social science research: principles, methods, and practices, Book 3, Open Access Textbooks. Retrieved December 31, 2013 from Coakes, SJ & Ong, C 2011 SPSS; analysis without anguish: version 18.0 for Windows. Wiley, Milton, Queensland Fink, A 2003, How to manage, analyse and interpret survey data, Sage, Thousand Oaks, CA. Sekaran, U & Bougie, R 2013, Research methods for business: a skill-building approach, 6 th edn. Wiley, Chichester MGMT617 Research Methods Section 9 10

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