Chapter 2. Chapter 2. Describing and Presenting a Distribution of Scores. © 2006 McGraw-Hill Higher Education. All rights reserved. Chapter Objectives. After completing this chapter, you should be able to Define all statistical terms that are presented.
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After completing this chapter, you should be able to
Nominal Scale: This scale refers to a classificatory approach, i.e., categorizing observations. Distinct characteristics must exist to categorize: gender, race essentially you can only be assigned one group. KEY: to distinguish one from another.
Ordinal Scale: This scale puts order into categories. It only ranks categories by ability, but there is no specific quantification between categories. It is only placement, e.g., judging a swimming race without a stopwatch, i.e., there is no quantitiy to determine the difference between ranks. KEY: placement without quantification.
Interval Scale: This scale adds equal intervals between observed categories. We know that 75 points is halfway between scores of 70 and 80 points on a scale. KEY: how much was the difference between 1st and 2nd place?
Ratio Scale: this scale has all the qualities of an interval scale with the added property of a true zero. Not all qualities can be assigned to a ratio scale. KEY: quality of measurement must represent a true zero.
The arithmetic average of a distribution of scores; most generally used measure of central tendency.
X = the mean (called X-bar)
= (Greek letter sigma) = “the sum of”
X = individual score
N = the total number of scores in distribution Mean Formula X = X
Table 2.3: X = 2644 = 88.1
Score that represents the exact middle of the distribution; the fiftieth percentile; the score that 50% of the scores are above and 50% of the scores are below.
Table 2.3: .50(30) = 15
Fifteenth and sixteenth scores are 88
P50 = 88
Score that occurs most frequently; may have more than one mode.
Least used measure of central tendency.
Not used for additional statistics.
Not affected by extreme scores.
Table 2.3: Mode = 88
Determined by subtracting the lowest score from the highest score; represents on the extreme scores.
1. Dependent on the two extreme scores.
2. Least useful measure of variability.
Formula: R = Hx - Lx
Table 2.3: R = 96 - 81 = 15
Sometimes called semiquartile range; is the spread of
middle 50% of the scores around the median. Extreme
scores will not affect the quartile deviation.
1. Uses the 75th and 25th percentiles; difference between
these two percentiles is referred to as the interquartile
2. Indicates the amount that needs to be added to, and
subtracted from, the median to include the middle
50% of the scores.
3. Usually not used in additional statistical calculations.
Q = quartile deviation
Q1 = 25th percentile or first quartile (P25) = score in which 25% of scores are below and 75% of scores are above
Q3 = 75th percentile or third quartile (P75) = score in which 75% of scores are below and 25% of scores are above
1. Arrange scores in ascending order.
2. Multiply N by .75 to find 75% of the distribution.
3. Count up from the bottom score to the number
determined in step 2. Approximation and interpolation
may be required.
Steps for Calculation of Q1
1. Multiply N by .25 to find 25% of the distribution.
2. Count up from the bottom score to the number
determined in step 1.
To Calculate Q
Substitute values in formula: Q = Q3 - Q1
Q1 = 25%
Q2 = 50%
Q3 = 75%
Q4 = 100%
Q2 - Q1 = range of scores below median
Q3 - Q2 = range of scores above median
1. Is the square root of the variance, which is the average of the squared deviations from the mean. Population variance is represented as F2 and the sample variance is represented as s2.
2. Is applicable to interval and ratio data, includes all
scores, and is the most reliable measure of variability.
3. Is used with the mean. In a normal distribution, one
standard deviation added to the mean and one standard
deviation subtracted from the mean includes the middle
68.26% of the scores.
4. With most data, a relatively small standard deviation
indicates that the group being tested has little
variability (performed homogeneously). A relatively
large standard deviation indicates the group has much
variability (performed heterogeneously).
5. Is used to perform other statistical calculations.
Symbols used to determine the standard deviation:
s = standard deviation X = individual score
X = mean N = number of scores
= sum of
d = deviation score (X - X)
1. Arrange scores into a series.
2. Find X2.
3. Square each of the scores and add to determine the X2.
4. Insert the values into the formula
NX2 - (X)2
s = N(N- 1)
X = 2644 N = 30
X2 = 233,398 s = 3.6
1. Arrange the scores into a series.
2. Calculate X.
3. Determine d and d2 for each score; calculate d2.
4. Insert the values into the formula
s = N - 1
X = 88.1 s = 3.6
d2 = 373.5
N = 30
S = 3.6
X = 88.1
88.1 + 3.6 = 91.7
88.1 - 3.6 = 84.5
In a normal distribution, 68.26% of the scores would fall between 84.5 and 91.7.
Based on the probability of a normal distribution, there is
an exact relationship between the standard deviation and
the proportion of area and scores under the curve.
1. 68.26% of the scores will fall between +1.0 and -1.0
2. 95.44% of the scores will fall between +2.00 and
-2.00 standard deviations.
3. 99.73% of the scores will fall between +3.0 and -3.00
4. Generally, scores will not exceed +3.0 and -3.0
standard deviations from the mean.
Class 1 Class 2
X = 32 X = 28
s = 2 s = 4
Figure 2.5 compares the spread of the two distributions.
Individual A in Class 1 completed 34 sit-ups and individual B completed 34 sit-ups in Class 2. Both individuals have the same score, but do not have the same relationship to their respective means and standard deviations. Figure 2.6 compares the individual performances.
1. Calculate the deviation of the score from the mean.
d = (X - X)
2. Calculate the number of standard deviation units the
score is from the mean (z-scores).
No. of standard deviation units from the mean = d
3. Use table 2.5 to determine where the percentile rank
of the score is on the curve. If negative value found in
step 1, the percentile rank will always be less than 50.
1. Range is the least desirable.
2. The quartile deviation is more meaningful than the
range, but it considers only the middle 50% of
3. The standard deviation considers every score, is the
most reliable, and is the most commonly used
measure of variability.
Percentile - a point in a distribution of scores below
which a given percentage of scores fall.
Examples - 60th percentile and 40 percentile
Percentile rank - percentage of the total scores that fall below a given score in a distribution; determined by beginning with the raw scores and calculating the
percentile ranks for the scores.
Frequency distribution – method for arranging the data in a
More convenient form.
Simple frequency distribution – all scores are listed in
Descending order and the number of times each individual
Scores occurs is indicated in a frequency column.
Table 2.6 shows a simple frequency distribution.
Sometimes more convenient to represent scores in a grouped
Step 1. Determine the range.
The highest score minus lowest score.
94 – 48 = 46
Step 2. Determine the number of class intervals.
Depends on the number of scores, the range of the scores, and the purpose of organizing the frequency table.
Generally it is best to have between 10 and 20 intervals.
Determine the size of the class interval (i).
Estimate if i can be found by dividing the range of scores
by the number of intervals wanted.
Example: if range of scores for a distribution = 54
54 = 3.6
Easier to work with whole numbers, so choice of 3 or 4 as
When i smaller than 10, generally best to use numbers
2, 3, 5, 7, and 9. May use even numbers, but midpoint of
of odd numbers will be whole number.
May also determine number of class intervals by dividing
the range by estimate of appropriate i.
Example: 54 = 18 or 54 = 11
Class intervals for tennis serve test:
46 = 3.06
With i = 3, there will be 16 intervals
See table 2.7.
Step 3. Determine the limits of the bottom class interval.
Usually begin bottom interval with a number that is
multiple of the interval size. May begin interval
with lowest score or make the lowest score the
midpoint of the interval.
Step 4. Construct the table.
Remaining intervals are formed by increasing each
interval by the size of i.
Note difference in the “apparent limits” and “real limits” of
Step 5. Tally the scores.
Step 6. Record the tallies under the column headed f and
sum the frequencies (f = N)
Note other columns in table 2.7. These columns are used to
calculate the measures of central tendency and variability.
With the exception of the mode, the definitions,
characteristics, and uses for these measures are the same.
Midpoint of the interval with the largest number of frequencies.
Calculated by adding ½ of i to the real lower limit (LL) of interval.
Mode in table 2.7 is
Mo = LL of interval + ½ (i)
= 71.5 + ½(3)
= 71.5 +1.5
Mo = 73
The calculation of the mean from the distribution in table 2.7 is
X = 73 + 3 -21
= 73 + 3(-.28)
= 73 - .84
X = 72.16
LL = the real lower limit of interval containing the percentile
% = the percentile you wish to determine
cfb = the cumulative frequency in the interval below the
interval of interest
fw = the frequency of scores in the interval of interest
The calculation of the median from the distribution in table 2.7 is
.50(75) = 37.5
P50 = 71.5 + 3 37.5 – 34
= 71.5 + 3 3.5
P50 =71.5 + 3(.35) = 72.55
Calculation of the range was described previously.
Quartile deviation and standard deviation will be covered
The calculation of Q from the distribution in table 2.7 is
.75(75) = 56.25 .25(75) = 18.75
Q3 = 80.5 + 3 56.25 – 56 Q1 = 62.5 + 3 18.75 – 16
= 80.5 + 3 .25 = 62.5 + 3 2.75
= 80.5 + .11 = 62.5 + 1.37
Q3 = 80.61 Q1 = 63.87
Q = Q3 – Q1
= 80.61 – 63.87
Q = 8.37
The calculation of s in the distribution in table 2.7 is
s = 3 973 - - 212
= 3 12.9733 – (.28)2
= 3 12.9733 - .0784
= 3 12.8949
s = 3 (3.59) = 10.77
1. Enable individuals to interpret data without reading
raw data or tables.
2. Different types of graphs are used.
Examples - histogram (column), frequency polygon (line),
pie chart, area, scatter, and pyramid
3. Standard guidelines should be used when constructing
See figures 2.7 and 2.8.
Provide method for comparing unlike scores; can obtain
an average score, or total score for unlike scores.
z-score - represents the number of standard deviations a
raw score deviated from the mean
z = X - X
Table 2.7- Tennis Serve Scores
Scores of 88 and 54; X = 72.2; s = 10.8
z = X - X
z = 88 - 72.2 = 15.8 z = 54 - 72.2 = -18.2
10.8 10.8 10.8 10.8
z = 1.46 z = -1.69
decimals, and may be positive or negative, many
testers do not use them.
Table 2.5 shows relationship of standard deviation units and percentile rank.
(this range includes plus and minus 3 standard deviations).
T-score = 50 + 10 (X - X) = 50 + 10z
Figure 2.9 shows the relationship of z-scores,
T-scores, and the normal curve.
Table 2.7 - Tennis Serve Scores
Scores of 88 and 54; X = 72.2; s = 10.8
T88 = 50 + 10(1.46) T54 = 50 + 10 (-1.69)
= 50 + 14.6 = 50 + (-16.9)
= 64.6 = 65 = 33.1 = 33
(T-scores are reported as whole numbers)
May convert raw scores in a distribution to T-scores
1. Number a column of T-scores from 20 to 80.
2. Place the mean of the distribution of the scores opposite the T-score of 50.
3. Divide the standard deviation of the distribution by ten. The standard deviation for the T-scale is 10, so each T-score from 0 to 100 is one-tenth of the standard deviation.
4. Add the value found in step 3 to the mean and each
subsequent number until you reach the T-score of 80.
5. Subtract the value found in step 3 from the mean and
each decreasing number until you reach the number 20.
6. Round off the scores to the nearest whole number.
*For some scores, lower scores are better (timed events).