Numeracy & Quantitative Methods: Numeracy for Professional Purposes Laura Lake

Descriptive statistics – conducting analysis on one variable at a time or univariate analysis. Common approaches to univariate analysis: Measures of distribution Measures of central tendency Measures of dispersion Recap: univariate analysis

Measures of dispersion: statistical measures that summarise the amount of spread or variation in the distribution of values in a variable. So, how values are spread within a distribution. There are a number of different measures (applicable to interval or ratio data): Range Standard deviation Variance Measures of dispersion

TypeDescription Range Difference between the highest (maximum) and lowest (minimum) value in the distribution of values VarianceThe measure of the spread. Standard deviationShows the relation that a set of data has to the mean of the sample data.

Range is simply the difference between the highest and lowest value in the distribution of values. Example: Weekly income of 10 people: Range is maximum income minus minimum income: = £150. Range £180£220£280£320£280£180£350£280£330£220

Of course, ordinal data can be ordered and so can give information on range. Example: Survey question – How useful did you find the book? Range is from very useful to very un-useful. Range – using ordinal data Very useful Very Un- useful Useful Un- useful Very Un- useful Useful Very useful Useful Un- useful

Inter quartile range (IQR) is another range measure but this time looks at the data in terms of quarters or percentiles. The range of data is divided into four equal percentiles or quarters (25%). Inter quartile range Min Max Q2 Median 50 th Percentile Q1 25 th percentile Q3 75 th percentile IQR Range

IQR is the range of the middle 50% of the data. Therefore, because it uses the middle 50%, it is not affected by outliers or extreme values. Outliers – variables that are the extreme lower or upper end of the distribution. They are atypical, infrequent observations. These will influence the mean (arithmetic). Why? 10 people record their height: 160, 162, 164, 166, 168, 170, 172, 174, 176 and 200 cm tall. With those values the mean is 171cm. 200cm is the outlier – take it out and the mean is 168cm. Inter quartile range

Where the mean is a measure of the centre of a group of numbers, the variance is the measure of the spread. It involves measuring the distance between each of the values and the mean. To calculate the variance : 1.calculate the mean 2. for each value in the distribution subtract the mean and then square the result (the squared difference) 3. calculate the average of those squared differences. Variance

= Sum of (observed value – mean score) 2 Total number of scores -1 The larger the variance value the further the observed values of the data set are dispersed from the mean. A variance value of zero means all observed values are the same as the mean. Variance

Standard deviation: how far on average each value is from the mean. Problem with variance: because the differences are squared, the units of variance are not the same as the units of the data. This can make interpretation of the results problematic. If the variance is square rooted, the units of variance then correspond to those of the data set. This square rooting of the variance is reported as the standard deviation. Standard deviation

So, in most disciplines, standard deviation is used more frequently than variance. Chart example of standard deviation. Standard deviation scores are used to generate standardised or z scores. o Standardised scores are individual values expressed in units of standard deviation from the mean. o Used to compare variables with different unit measures. Standard deviation

Standard deviation = The square root of the variance. As it is square rooted the results correspond to the original data units. E.g. if the variable is height recorded in cm then the standard deviation can be interpreted as cm. Standard deviation

Appropriate descriptive statistics: summary Level of measurement Univariate analysis Nominal Frequency table: count, %, valid %, cumulative %. Measure of central tendency: mode Measure of dispersion: no measure. Ordinal Frequency table: count, %, valid %, cumulative %. Measure of central tendency: mode, median Measure of dispersion: no measure. Interval/Ratio Frequency table: count, %, valid %, cumulative %. Measure of central tendency: mode, median, mean Measure of dispersion: range, variance, standard deviation

Bryman, A. (2008) Social Research Methods. 3rd Ed. Oxford: Oxford University Press. David, M. and Sutton, C. (2011) Social Research : An Introduction. 2nd ed. London: Sage. References

This resource was created by the University of Plymouth, Learning from WOeRk project. This project is funded by HEFCE as part of the HEA/JISC OER release programme.Learning from WOeRk This resource is licensed under the terms of the Attribution-Non-Commercial-Share Alike 2.0 UK: England & Wales license ( The resource, where specified below, contains other 3 rd party materials under their own licenses. The licenses and attributions are outlined below: 1.The name of the University of Plymouth and its logos are unregistered trade marks of the University. The University reserves all rights to these items beyond their inclusion in these CC resources. 2.The JISC logo, the and the logo of the Higher Education Academy are licensed under the terms of the Creative Commons Attribution -non-commercial-No Derivative Works 2.0 UK England & Wales license. All reproductions must comply with the terms of that license. Author Laura Lake InstituteUniversity of Plymouth Title Numeracy & Quantitative Methods Numeracy for Professional Purposes Description Basic Descriptive Statistics: Introduction Date Created May 2011 Educational Level Level 4 Keywords Learning from WOeRK Work Based Learning WBL Continuous Professional Development CPD Research UKOER LFWOER Measures of dispersion, range, variance, standard deviation. Back page originally developed by the OER phase 1 C-Change project ©University of Plymouth, 2010, some rights reserved