Understanding and Interpreting Statistics in Assessments Clare Trott and Hilary Maddocks

This Session Why use statistics in assessments? “Averages”, Standard Deviation, variance, Standard Error Normal distribution, confidence intervals Scales Overlapping confidence intervals

Why use statistics in assessments? What are the assumptions made?

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AVERAGE MEANMEDIANMODE Which is better? When? What is Standard Deviation? Variance? Standard Error? Central Tendency Spread

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MODE MOST OFTEN D E MEDIAN MEDIAN M ake Everyone Add Numbers (and) Share Means

Standard Deviation 2, 3, 6, 9, 10 Mean = 6, SD = , 2, 6, 10, 10 Mean = 6, SD = 3.58 Measures the average amount by which all the data values deviate from the mean Measured in the same units as the data

Variance and Standard Deviation Variance =

Standard Error This is the variance per person

Normal Distribution What is Normal Distribution? Why is it useful? Confidence Intervals What are Confidence Intervals? Why are they important?

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Number of standard deviations from the mean Normal Distribution

Confidence Intervals The wider the range the more confident we can be that the true score lies in this range TRUE SCORE Due to inherent error in measurement it is better to quote a 95% confidence interval 99% Confidence Interval 95% Confidence Interval C I

Confidence Intervals % Confidence Interval 95% Confidence Interval 99% Confidence Interval

True Score True score lies inside CI 95% of occasions 1 in 20 (5%) will not include the true score 95% Confidence Intervals

Scales What scales are used in reporting? How are they defined? Why are standardised scores preferred?

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Very lowlow Low average average High average highVery high Scaled scores Standardised scores Percentiles

standar dised percent ile scaled 130 and above 98th>16+ 3SDWithin top 2% Very high SDAbove 91% high SDAbove 75% High average MeanAbove 25% average SDAbove 16% Low average SDAbove 10% Below average Below 70 Below 2< 4-3SDLowest 2% Very low Simplified Table

Scale to Standardised 1 to 5 ratio 10 scaled 100 standardised 9 scaled 95 standardised 11 scaled 105 standardised 15 scaled 125 standardised 6 scaled 80 standardised

Very lowlow Low average average High average highVery high Standardised scores against standard deviations -1sd -2sd -3sd 1sd 2sd 3sd mean

Very lowlowaverage High average high Very high Low average Percentiles against standard deviations -1sd -3sd -2sd 3sd 2sd 1sd mean

Very lowlowaverage High average high Very high Low average Scaled scores against standard deviations mean 3sd -2sd -3sd 1sd-1sd 2sd

Differences in Class Intervals Suppose we have the class intervals for two tests which could be linked, and we wish to find whether there is a significant difference between the two sets. Test 1 95% Confidence Interval 102 ± 15.8, standard error 2.96 Test 2 95% Confidence Interval 118 ± 23, standard error There appears to be no significant difference as there is a distinct overlap.

H 0 : There is no significant difference in the two Confidence Intervals (the new confidence interval contains zero) H 1 : There is a significant difference in the two Confidence Intervals (the new CI does not contain zero =(118 – 102) Formula Difference in scores = 16 ± 14 New CI This does not contain zero so we reject H 0 and so there is a significant difference in the two tests.

Test 1 95% Confidence Interval 95 ± 6, standard error 3.06 Test 2 95% Confidence Interval 106 ± 10, standard error There appears to be no significant difference as there is a distinct overlap.

H 0 : There is no significant difference in the two Confidence Intervals (the new confidence interval contains zero) H 1 : There is a significant difference in the two Confidence Intervals (the new CI does not contain zero =(106 – 95) Difference in scores = 11 ± 11.6 New CI This does contain zero so we accept H 0 and so there is no significant difference in the two tests.