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Basic Statistics
Module 6
Activity 4
© 2016 by Northern Arizona University. Basic Statistics for the AE E-Teacher Program, sponsored by the U.S. Department of State and administered by FHI 360. This work is licensed under the
Creative Commons Attribution 4.0 License, except where noted. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
This is a program of the U.S. Department of State administered by FHI 360 and delivered by Northern Arizona University.
Slide 1 of 31
Overview
In this presentation, you will learn about basic statistics. You will
lean about the following topics:
1. Measures of central tendency
2. Measures of variability
3. Distribution
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FHI 360 and delivered by Northern Arizona University.
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Slide 2 of 31
Measures of central
tendency
© 2016 by Northern Arizona University. Basic Statistics for the AE E-Teacher Program, sponsored by the U.S. Department of State and administered by FHI 360. This work is licensed under the
Creative Commons Attribution 4.0 License, except where noted. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
This is a program of the U.S. Department of State administered by FHI 360 and delivered by Northern Arizona University.
Slide 3 of 31
Measures of central tendency
A measure of central tendency gives you information about the
average or common score in a group of scores. There are three
measures of central tendency:
Mean
Median
Mode
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Slide 4 of 31
Mean
The mean is also known as the arithmetic average. To get the
mean of a group of scores, you add all of the scores and divide
by the number of scores.
For example, imagine that you give 10 tests. The scores are: 9,
8, 8, 9, 10, 7, 7, 4, 5, 8. To get the mean you would add all of the
scores and divide by 10.
9+8+8+9+10+7+7+4+5+8 = 75
75 / 10 = 7.5
The mean is 7.5
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Slide 5 of 31
Median
The median is the midpoint of the scores. To get the median, you
arrange the scores in order from lowest to highest and find the
middle score. If you have an even number of scores, the median
is the midpoint between the two halves.
For example, let’s find the median of our earlier scores: 9, 8, 8, 9,
10, 7, 7, 4, 5, 8.
First, we re-arrange them: 4, 5, 7, 7, 8, 8, 8, 9, 9, 10
There’s an even number of scores, so our median is between the
8 and the 8. So, the median is 8.
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Slide 6 of 31
Mode
The mode is the most frequent score.
For example, let’s find the mode of our earlier scores: 9, 8, 8, 9,
10, 7, 7, 4, 5, 8.
The most frequent score is 8, so the mode is 8.
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Slide 7 of 31
Measures of central tendency
Mean, median, or mode?
Calculating the mean takes into account each score, so really
high or really low scores will affect the mean.
Median is a counting average, so really high or really low scores
will not affect it.
Mode is simply the most frequent score, so it is the least reliable
measure of central tendency.
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Slide 8 of 31
Measures of central tendency
Why would you use the mean, median, or mode?
These three measures are quick and easy calculations to see
how a student group performed on an assessment. They tell you
information about the overall group of students.
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Slide 9 of 31
Measures of variability
© 2016 by Northern Arizona University. Basic Statistics for the AE E-Teacher Program, sponsored by the U.S. Department of State and administered by FHI 360. This work is licensed under the
Creative Commons Attribution 4.0 License, except where noted. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
This is a program of the U.S. Department of State administered by FHI 360 and delivered by Northern Arizona University.
Slide 10 of 31
Measures of variability
A measure of variability gives you information about how much
scores spread out from the measure of central tendency. We will
look at two measures of variability:
Range
Standard deviation
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Slide 11 of 31
Range
Range tells you the distance from the highest score to the lowest
score. You get it by subtracting the lowest score from the highest
score.
For example, let’s go back to our earlier scores: 9, 8, 8, 9, 10, 7,
7, 4, 5, 8.
The highest score is 10, and the lowest score is 4.
10 – 4 = 6.
The range is 6.
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Slide 12 of 31
Standard Deviation
The standard deviation, or sd, is a measure of the variability of
scores. To find the sd, there are five steps:
1. Add all of the scores together. Take the result and multiple it
by itself. Divide this number by the number of scores.
2. Multiple each individual score by itself. Add these numbers
together.
3. Subtract the number from step 1 from the number in step 2.
4. Divide the number from step three by the number of scores
minus one.
5. Find the square root of the number from step 4.
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Slide 13 of 31
Standard deviation
Let’s calculate the sd for our earlier set of scores: 9, 8, 8, 9, 10, 7,
7, 4, 5, 8.
Step 1: Add all of the scores together. Take the result and
multiple it by itself. Divide this number by the number of scores.
9+8+8+9+10+7+7+4+5+8 = 75
75*75 = 5,625
5,625 / 10 = 562.5
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Slide 14 of 31
Standard deviation
Step 2: Multiple each individual score by itself. Add these numbers together.
9*9 = 81
8*8 = 64
8*8 = 64
9*9 = 81
10*10 = 100
7*7 = 49
7*7 = 49
4*4 = 16
5*5 = 25
8*8 = 64
Sum = 593
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Slide 15 of 31
Standard deviation
Step 3: Subtract the number from step 1 from the number in step
2.
593 – 562.5 = 30.5
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Slide 16 of 31
Standard deviation
Step 4: Divide the number from step three by the number of
scores minus one.
30.5 / 9 = 3.39
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Slide 17 of 31
Standard deviation
Step 5: Find the square root of the number from step 4.
Square root of 3.39 = 1.84
The standard deviation of these scores: 9, 8, 8, 9, 10, 7, 7, 4, 5, 8
is 1.84.
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Slide 18 of 31
Measures of variability
The range of scores is a very simple measure of variability. The
standard deviation is a more informative measure. The sd tells
you how much, on average, scores are different from the mean.
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Slide 19 of 31
Measures of variability
A small range or sd tells you that the scores do not vary a lot. A
large range of sd tells you that the scores are very different.
For a classroom achievement test, we hope that the average
score is high with a small sd. A small sd would indicate that
students are performing similarly.
For a proficiency placement test, we expect to see a variety of
scores. For this type of test, a larger sd would be expected.
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Slide 20 of 31
Measures of variability
Why do we use the range and standard deviation?
The mean median and mode tell you how the students did
generally. The range and standard deviation give you a better
understanding of how all of your students did. The range can
show the difference between the high and low scoring students.
The standard deviation helps you understand the variation
between your students.
This is a program of the U.S. Department of State administered by
FHI 360 and delivered by Northern Arizona University.
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Slide 21 of 31
Distribution
© 2016 by Northern Arizona University. Basic Statistics for the AE E-Teacher Program, sponsored by the U.S. Department of State and administered by FHI 360. This work is licensed under the
Creative Commons Attribution 4.0 License, except where noted. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
This is a program of the U.S. Department of State administered by FHI 360 and delivered by Northern Arizona University.
Slide 22 of 31
Distribution
The distribution is the shape of the scores. It is easiest to
understand visually. Look at the image below. This may look
familiar to you.
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Slide 23 of 31
Distribution
Across the bottom, you can see
the grades on a test, from 50 to
100. Up the left side, you can see
the number of students. You can
see that 20 students received an
80 on the test, but only 3
received a 100.
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Slide 24 of 31
Distribution
We call this a normal distribution
because if you draw a vertical
line through the middle, the left
side will be the same as the right
side.
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Slide 25 of 31
Distribution
There are many different ways that scores can be distributed.
The image on the left is positively skewed and the image on the
right is negatively skewed.
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Slide 26 of 31
Distribution
The distribution of scores will depend on the test that we are
giving.
A norm-referenced test should have a relatively normal
distribution.
A criterion-referenced test will have either a positively or
negatively skewed distribution. If it is a pre-test where students
do not know the material, it should be positively skewed. If it is an
achievement test, it should be negatively skewed.
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Slide 27 of 31
Distribution
Achievement test:
negatively skewed, most
of the scores are high.
Pre-test: positively
skewed, most of the
scores are low.
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Slide 28 of 31
Key Terms & Concepts
Review:
Mean
Median
Mode
Standard deviation
Normal distribution
This is a program of the U.S. Department of State administered by
FHI 360 and delivered by Northern Arizona University.
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Slide 29 of 31
Next Steps
Now you should read the Reliability PowerPoint presentation.
This is activity #5.
This is a program of the U.S. Department of State administered by
FHI 360 and delivered by Northern Arizona University.
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Slide 30 of 31
Sources
Carr, N. T. (2011). Designing and analyzing language tests.
Oxford: Oxford University Press.
Miller, M. D., Linn, R. L., & Gronlund, N. E. (2009). Measurement
and assessment in teaching. New Jersey: Pearson.
This is a program of the U.S. Department of State administered by
FHI 360 and delivered by Northern Arizona University.
PARTNER
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Slide 31 of 31