statistics for interpreting test scores
DESCRIPTION
The slides touche briefly statistical procedures employed for interpreting test results. The file is especially useful for ESL teachersTRANSCRIPT
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Kinds of Statistics
Purpose Target Characteristics
DescriptiveStatistics Summarizing,
Describing Sample Statistic
Inferential Statistics
Analyzing, Generalization Population Parameter
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Tabulation of Data
Ungrouped Ungrouped Data Data
Grouped Data Grouped Data
151599
161613131111101088
1212
1616151513131212111110109988
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Frequency Distribution
• Graphic description of how many times a score or group of scores occurs in a sample
• Common symbol is “f”
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Absolute Frequency
Score(X) Score(X) Frequency(f) Frequency(f) 1616151513131212111110109988
1111225544441122
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Frequency distribution
Score (X) Score (X) Absolute Absolute FrequencyFrequency
Relative Relative FrequencyFrequency
1616151513131212111110109988
1111225544441122
0.050.050.050.050.100.100.250.250.200.200.200.200.050.050.100.10
Σx = 94 N= 20N= 20 1.001.00
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Frequency distributionScore Score
(X) (X) Absolute Absolute
FrequencyFrequencyRelative Relative
FrequencyFrequency PercentagePercentage
1616151513131212111110109988
1111225544441122
0.050.050.050.050.100.100.250.250.200.200.200.200.050.050.100.10
0.05x100=50.05x100=50.05x100=50.05x100=50.10x100=10.10x100=1000.25x100=20.25x100=2550.20x100=20.20x100=2000.20x100=20.20x100=2000.05x100=50.05x100=50.10x100=10.10x100=100
Σx = 94 N= 20N= 20 1.001.00 100100
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Cumulative Frequency
• Cumulative frequency distribution is a graphic depiction of the how many times groups of scores appear in a sample
• Common symbol is “cf”
• “cf “ is used to compute percentile scores
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Cumulative frequency
XX ff cfcf
1616
1515
1313
1212
1111
1010
99
8 8
11
11
22
55
44
44
11
2 2
2020
1919
1818
1616
1111
77
33
2 2
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Percentile score showing relative standing in a distribution showing what percentage of scores are higher and lower than
a certain score.Percentile computation cf
P(percentile) = (100) ----- N
N= number of scores Cf =cumulative frequency Cf 16 P= (100)----- = (100)------ = 80 N 20 Cf 20 P = (100)-----= (100) ------ = 100 N 20
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Bar graph
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Frequency Polygon
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Measures of Central Tendency
• Mean: arithmetic average of all scores in a distribution
• Median: the point at which exactly half of the scores in a distribution are below & half are above
• Mode: most frequently occurring score(s)
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Measures of central tendency1. Mean / arithmetic average _ Σx X = ------- Σx = sum of all scores
N N = number of scores
Example:Example:13 + 14 + 15 + 16 + 17 = 7513 + 14 + 15 + 16 + 17 = 75
ΣΣx = 75x = 75 N= 5N= 5
_ _ ΣΣx 75x 75 X = -------X = ------- = = -------= 15-------= 15 NN 5 5
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2. ModeXX f f
1616
1515
131312
1111
1010
99
88
11
11
225
44
44
11
2 2
1. Odd number 13, 15, 16, 17, 19
2. Even number
12, 13, 12, 13, 15, 1715, 17, 18, 19, 18, 19 = = 1616
3.Median:
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Measures of Variability
• These describe the spread, or dispersion, of scores in a distribution
• These measures describe the nature & extent to which scores vary
• Three most commonly used measures are:
1. Range2. Variance
3. Standard Deviation
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Measures of variability1. Range
13, 15, 16, 17, 19 19-13= 6
2.2. Variance (V)Variance (V)
V =------------- N -1
3. standard deviation3. standard deviation
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XX (X - )(X - ) ΣΣ(X - )(X - )22
19191818171716161515141413 13
+ 3+ 3+2+2+1+100-1-1-2-2-3 -3
9944110011449 9
112112 00 2828
ExampleExample
Σx2 28 (V( = ---------= --------=4.6 N-1 7-1
S = 2.14
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Normal /bell-shaped curve
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Properties of Normal /bell-shaped curve
• It is a symmetrical distribution• Most of the scores tend to occur near the center– while more extreme scores on either side of the center
become increasingly rare. – As the distance from the center increases, the
frequency of scores decreases. • The mean, median, and mode are the same.
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Normal Probability Curve
• Describes an expected distribution of scores in a population or sample
• More than 2/3 of scores cluster in the middle of the curve
• Scores that are extremely high or low are sometimes called outliers
• Shaped like a bell
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Normal /bell-shaped curve
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Symmetrical vs. asymmetrical d distribution
• In a symmetrical distribution– the part of the histogram on the left side of the fold
would be the mirror image of the part on the right side of the fold.
• In a asymmetrical, distribution• the two sides will not be mirror images of each other.
True symmetric distributions include what we will later call the normal distribution.
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• A asymmetric distribution is either Positively or negatively skewed. – In a positively skewed distribution the scores
cluster toward the lower end of the scale (that is, the smaller numbers) with increasingly fewer scores at the upper end of the scale (that is, the larger numbers).
– With a negatively skewed distribution, most of the scores occur toward the upper end of the scale while increasingly fewer scores occur toward the lower end.
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Kurtosis Distribution• Mesokurtic distribution : with normal
distribution of scores• Leptokurtic distribution: packed, with low
variability of scores • Platykurtic Distribution: flat, with high
variability of scores
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Examples based on the curveAdults intelligence = 100 SD= 15
Mean +1 SD = 68%100 + 15 = 85 -115
34 percent = 100--115 34 percent = 85 -- 100
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Mean + 2 SD = 94%100 + (2 X 15) = 70 --130
115 + 15 = 13013 % = 115 – 130
85 – 15 = 7013 % = 70 –85
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Standard ScoresTo compare scores on different measurement scales
I. Z-Scores: the commonest score
Z-score propertiesHow many scores above/below the meanThe mean being set at zeroThe SD being set at one
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Standard Scores
II. T-Score: A standard score whose distribution has a mean of 50 and a standard deviation of 10.
Advantages of T-scoreEnabling us to work with whole numbersAvoiding describing subjects’ performances
with negative numbers
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An example:
_
25- 20 = ---------- = +1 5
X - 30-35 Z = ---------- = -------- = -1 SD 5Ali’s Z-score : +1
Better than 84% of the classAhmad’s Z-score: --1
Better than only 16% 0f the class
Student X SD
Ali 25 5 20
Ahmad 30 5 35
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An example:
_
X – 76- 54 Z = --------- = ---------- = +1.1 SD 20
X - 82- 72 Z = ---------- = -------- = 0.67 SD 15So, using standard scores, Ali did better than Ahmad because
Ali’s mark was more standard deviations above the class mean than Ahmad’s score was above his own class mean.
Student X SD
Ali 76 20 54
Ahmad 82 15 72
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Negative SkewNegative Skew Positive skewPositive skew
Test items were easy.
Testees performed well.
The score are far from zero.
Test items were difficult.
Testees performed poorly.
The scores are near zero.
Skewed Distribution
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The Coefficient of Correlation, r
The Coefficient of Correlation (r) is a measure of the strength of the relationship between two variables. It requires interval or ratio-scaled data.
It can range from -1.00 to 1.00.Values of -1.00 or 1.00 indicate perfect and
strong correlation.Values close to 0.0 indicate weak correlation.Negative values indicate an inverse relationship
and positive values indicate a direct relationship.
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Perfect Correlation
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Correlation Coefficient - Interpretation
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CorrelationGo-togetherness of variablesGo-togetherness of variablesNo cause-effect relationshipNo cause-effect relationshipBetween -1 and +1Between -1 and +1
Types of correlation:1.Pearson Product-moment
a. Used for interval data
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Correlation Coefficient - Formula
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2. Spearman rank-order, rhoa. used for ranked or ordinal data
StudentsStudents Teacher 1Teacher 1 Teacher 2Teacher 2 DD DD22
AA
BB
CC
DD
E E
11
22
33
44
5 5
55
44
33
22
1 1
44
22
00
22
4 4
1616
44
00
44
16 16
55 1515 1515 1212 4040
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6 (40) 240 P = 1-- ––––––– = ––––––––
5 (25-1) 120
1 – 2 = -1
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Standard error of measurement (SEM)True score= Raw score – Random errors
How to estimate
X = scoren = number of items
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An example:X= 79 n = 100
X (n- X) 79 (100-79) SEMX = –––––––– = –––––––––––= 4
n – 1 100-1
79 + 1SEM = 75-83 68% 79 + 2SEM = 71-87 95%
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Point-biserial correlation:
rpbi = point-biserial correlation coefficient
Mp = whole-test mean for students answering item correctly (i.e., those coded as 1s)
Mq = whole-test mean for students answering item incorrectly (i.e., those coded as 0s)
St = standard deviation for whole test
p = proportion of students answering correctly (i.e., those coded as 1s)
q = proportion of students answering incorrectly (i.e., those coded as 0s)
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As another example, where
the whole-test mean for Ss answering correctly is
30;
the whole-test mean for Ss answering incorrectly is
45;
the standard deviation for the whole test is still
8.29;
the proportion of Ss answering correctly is still .50;
and the proportion answering incorrectly is
still .50.
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Types of Correlation coefficientsType of Correlation
CoefficientTypes of Scales
Pearson product-moment
Both scales interval (or ratio)
Spearman rank-order
Both scales ordinal
Phi Both scales are naturally dichotomous (nominal)
BiserialOne scale artificially dichotomous
(nominal), one scale interval (or ratio)
Point-biserial One scale naturally dichotomous (nominal), one scale interval (or ratio)
Gamma One scale nominal, one scale ordinal
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Correction for guessing
A student has taken a test of 100 items. As he has no knowledge, he takes choice A and marks it for all the items. His true / corrected score is
75 Corrected score= 25--
------------ 4—1
25-25= 0
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Some other factors affecting a person’s score:
Practice effectCoaching effectCeiling effectTest compromiseTest MethodTest Taker’s characteristics
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In a normal distribution, what percentage of scores fall between the mean and one standard deviation? a. 35% b. 50%
c. 68% d. 95%
A test has been given to 100 students. Twenty students have obtained the score of 50. What is the percentage of this score?
a. 10 b. 15 c. 20 d. 30
f 20P x= ------ X 100 = ------= 0.20 X 100 = 20 N 100
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A test is administered to 100 students. The cumulative frequency of the score of 50 is 40. How many students have scores below the score of 50?
a. about 25 b. about 20 c. about 40 d. about 30
N=100 cf=40 cf 40 P= (100)----- = 100 x ------ = 40 N 100
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In a test eight of the students obtained a score of 85. This score has the highest frequency. What is the label used for this score?a. Mean b. Mode c. Median d. Range
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Fry’s Readability Graph
Directions for Use Randomly select three 100-word passages from a book or an
article. Plot the average number of syllables and the average number of
sentences per 100 words on the graph to determine the grade level of the material.
Choose more passages per book if great variability is observed and conclude that the book has uneven readability.
Few books will fall into the solid black area, but when they do, grade level scores are invalid.
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Additional Directions• Randomly select three sample passages and count exactly 100
words beginning with the beginning of a sentence. Don't count numbers. Do count proper nouns.
• Count the number of sentences in the hundred words, estimating length of the fraction of the last sentence to the nearest 1/10th.
• Count the total number of syllables in the 100-word passage. • Enter graph with average sentence length and number of
syllables; plot dot where the two lines intersect. Area where dot is plotted will give you the approximate grade level.
• If a great deal of variability is found, putting more sample counts into the average is desirable.
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