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
MEDIAN
MEAN
MODE
Central Tendency
AVERAGE
Spread
Which is better? When?
What is
• Standard Deviation?
• Variance?
• Standard Error?
Feedback
MODE
MOST
OFTEN
D
E
MEDIAN
MED IAN
Means Make
Everyone
Add
Numbers
(and)
Share
Standard Deviation
2, 3, 6, 9, 10
Mean = 6, SD = 3.16
2, 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 = 𝜎 2
Σ 𝑥−𝑥
=
𝑛
2
Mean (𝑥 )
Standard deviation
=
σ=
Σ(𝑥−𝑥)2
𝑛
Standard Error
This is the variance per person
𝜎2
𝑆𝐸 =
𝑛
Normal Distribution
Confidence Intervals
• What is
Normal Distribution?
• What are
Confidence Intervals?
• Why is it useful?
• Why are they
important?
Feedback
Normal Distribution
Number of standard deviations from the mean
Confidence Intervals
TRUE SCORE
99%
95%
Confidence Confidence
Interval
Interval
CI
The wider the range the
more confident we can
be that the true score
lies in this range
Due to inherent error in
measurement it is better
to quote a 95%
confidence interval
Confidence Intervals
1.645
-1.645
-1.96
1.96
-2.575
2.575
90% Confidence Interval
95% Confidence Interval
99% Confidence Interval
95% Confidence Intervals
True Score
• True score lies inside CI 95% of occasions
• 1 in 20 (5%) will not include the true score
Scales
• What scales are used in reporting?
• How are they defined?
• Why are standardised scores preferred?
Feedback
Scaled scores
4
6
8
10
12
14
16
Percentiles
2
10
25
50
75
90
98
120
130
Standardised scores
70
Very low
80
low
90
Low
average
100
average
110
High
average
high
Very high
Simplified Table
standar
dised
percent
ile
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
Scale to Standardised
•
•
•
•
1 to 5 ratio
10 scaled
9 scaled
11 scaled
• 15 scaled
• 6 scaled
100 standardised
95 standardised
105 standardised
125 standardised
80 standardised
Standardised scores against standard deviations
mean
-1sd
1sd
-2sd
2sd
3sd
-3sd
70
Very low
80
low
90
Low
average
100
average
110
High
average
120
high
130
Very high
Percentiles against standard deviations
mean
-1sd
1sd
-2sd
2sd
-3sd
3sd
2
Very low
10
low
25
Low
average
50
average
75
High
average
90
high
98
Very
high
Scaled scores against standard deviations
mean
-1sd
1sd
-2sd
2sd
-3sd
3sd
4
Very low
6
low
8
Low
average
10
average
12
High
average
14
high
16
Very
high
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 6.63
86.2
102
105
117.8
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
Formula
Difference in scores ±1.96 𝑆𝐸12 + 𝑆𝐸22
=(118 – 102) ±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 1
95% Confidence Interval 95 ± 6, standard error 3.06
88
95
102
Test 2
95% Confidence Interval 106 ± 10, standard error 5.102
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
Difference in scores
=(106 – 95)
±1.96 𝑆𝐸12 + 𝑆𝐸22
±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.