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University of Durham D Dr Robert Coe University of Durham School of Education Tel: (+44 / 0) 191 334 4184 Fax: (+44 / 0) 191 334 4180 E-mail: r.j.coe@dur.ac.uk http://www.dur.ac.uk/r.j.coe Summarising and Presenting Data Doctor of Education (EdD) Analysing, Interpreting and Using Educational Research (Research Methodology)

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© 2005 Robert Coe, University of Durham 2 Look at the data Test scores

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© 2005 Robert Coe, University of Durham 3 Histogram

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© 2005 Robert Coe, University of Durham 4 Stem and leaf plot Average residual Stem-and-Leaf Plot for NAMED= named Frequency Stem & Leaf 1.00 -3. 5 1.00 -3. 0 5.00 -2. 55679 3.00 -2. 013 9.00 -1. 666677899 10.00 -1. 0001112344 14.00 -0. 55555666788999 10.00 -0. 0011223344 6.00 0. 001233 1.00 0. 5 1.00 Extremes (>=1.7) Stem width: 1.000 Each leaf: 1 case(s)

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© 2005 Robert Coe, University of Durham 5 Box and whisker Median Highest value Lowest value Upper quartile Lower quartile Outlier

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© 2005 Robert Coe, University of Durham 6 Averages: measures of ‘central tendency’ Mean: (X 1 + X 2 + … + X n ) / n For ‘well behaved’ data (symmetric, Normal, interval property), the mean is most efficient estimator (most accurate for given sample size) Fits well in statistical calculations Median: arrange in order, median is middle value Not dependent on interval property More robust to outliers Less ‘efficient’ than the mean

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© 2005 Robert Coe, University of Durham 7 Mode: most frequent value Suitable for non-ordered data Otherwise seldom used Trimmed means: chop off (e.g.) top 10% and bottom 10%, calculate mean of remaining Almost as efficient as mean Far more robust Seldom used (Wilcox, 1998)

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© 2005 Robert Coe, University of Durham 8 Which average? Grades achieved in GCSE Maths: Mode = C Mean = 4.3 Median = D Coded as: 0 1 2 3 4 5 6 7 8

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© 2005 Robert Coe, University of Durham 9 Averages (graphically) Income Mean : Balancing point Median : Divides equally Mode : Highest point 12

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© 2005 Robert Coe, University of Durham 10 Exercise 1. On each plot, show: Mean Median

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© 2005 Robert Coe, University of Durham 11 Standard deviation Measure of spread, ‘range’ 3, 3, 4, 4, 4, 4, 4, 4, 5, 5 3, 3, 3, 3, 4, 4, 5, 5, 5, 5 2, 2, 3, 3, 4, 5, 5, 5, 5, 6 mean 4 4 4 Standard deviation (SD) = average distance from mean (actually uses distance 2 ) Hence, further away points influence more range 2 2 4 SD 0.6 0.9 1.3

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© 2005 Robert Coe, University of Durham 12 Standard deviation (graphically) Standard Deviation Mean ± 1 Standard Deviation 1/61/6 1/61/6 1/61/6 1/61/6 1/61/6 1/61/6 2 / 3 of the population Mean ± 2 Standard Deviations 95% of the population

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© 2005 Robert Coe, University of Durham 13 Exercise 2. On each plot, show: Standard deviation

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