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?
Feedback
AVERAGE
MEAN
MEDIAN
MODE
Which is better? When?
What is • Standard Deviation? • Variance? • Standard Error?
Central Tendency
Spread
Feedback
MODEMOSTOFTENDE MEDIAN
MED IAN
Make
Everyone
Add
Numbers
(and)Share
Means
Standard Deviation2, 3, 6, 9, 10Mean = 6, SD = 3.16
2, 2, 6, 10, 10Mean = 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
Σ (𝑥−𝑥 )2
𝑛Variance =
Mean ()
Standard deviation = σ =𝜎 2=¿
Standard Error
This is the variance per person
𝑆𝐸=𝜎2
𝑛
Normal Distribution
• What is Normal Distribution?
• Why is it useful?
Confidence Intervals
• What are Confidence Intervals?
• Why are they important?
Feedback
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
IntervalC I
Confidence Intervals
-1.645 1.645
1.96-1.96
2.575-2.575
90% 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?
Feedback
Very low low Low average average High
average high Very high
100 130110 120908070
50 9875 902510 2
10 1612 14864
Scaled scores
Standardised scores
Percentiles
standardised
percentile
scaled
130 and above
98th >16 + 3SD Within top 2%
Very high
120-129 91-97 14-15 + 2SD Above 91%
high
110-119 75-90 12-13 + 1SD Above 75%
High average
90-109 25-74 8-11 Mean Above 25%
average
80-89 10-24 6-7 -1SD Above 16%
Low average
70-79 2-9 4-5 -2SD Above 10%
Below average
Below 70 Below 2 < 4 -3SD Lowest 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
100 130110 120908070
Very low low Low average average High
average high Very high
Standardised scores against standard deviations
-1sd
-2sd
-3sd
1sd
2sd
3sd
mean
50 9875 902510 2
Very low low average High average high Very
highLow average
Percentiles against standard deviations
-1sd
-3sd
-2sd
3sd
2sd
1sd
mean
10 1612 14864
Very low low average High average high Very
highLow average
Scaled scores against standard deviationsmean
3sd
-2sd
-3sd
1sd-1sd
2sd
Differences in Class IntervalsSuppose 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 195% Confidence Interval 102 ± 15.8, standard error 2.96
Test 295% Confidence Interval 118 ± 23, standard error 6.63
86.2 102 117.8 105 118 131
There appears to be no significant difference as there is a distinct overlap.
H0 : There is no significant difference in the two Confidence Intervals (the new confidence interval contains zero)
H1 : There is a significant difference in the two Confidence Intervals (the new CI does not contain zero
=(118 – 102)
Formula ±1.96√𝑆𝐸 12+𝑆𝐸2
2Difference in scores
±1.96√2.962+6.632
= 16 ± 14
New CI 2 16 30
This does not contain zero so we reject H0 and so there is a significant difference in the two tests.
Test 195% Confidence Interval 95 ± 6, standard error 3.06
Test 295% Confidence Interval 106 ± 10, standard error 5.102
88 95 102
96 106 116
There appears to be no significant difference as there is a distinct overlap.
H0 : There is no significant difference in the two Confidence Intervals (the new confidence interval contains zero)
H1 : There is a significant difference in the two Confidence Intervals (the new CI does not contain zero
=(106 – 95)
±1.96√𝑆𝐸 12+𝑆𝐸 2
2Difference in scores
±1.96√3.062+5.1022
= 11 ± 11.6
New CI -0.6 11 22.6
This does contain zero so we accept H0 and so there is no significant difference in the two tests.