University of Durham D Dr Robert Coe University of Durham School of Education Tel: (+44 / 0) Fax: (+44 / 0) Summarising and Presenting Data Doctor of Education (EdD) Analysing, Interpreting and Using Educational Research (Research Methodology)

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

© 2005 Robert Coe, University of Durham 3 Histogram

© 2005 Robert Coe, University of Durham 4 Stem and leaf plot Average residual Stem-and-Leaf Plot for NAMED= named Frequency Stem & Leaf Extremes (>=1.7) Stem width: Each leaf: 1 case(s)

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

© 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

© 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)

© 2005 Robert Coe, University of Durham 8 Which average? Grades achieved in GCSE Maths: Mode = C Mean = 4.3 Median = D Coded as:

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

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

© 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 Standard deviation (SD) = average distance from mean (actually uses distance 2 ) Hence, further away points influence more range SD

© 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

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