Chapter Three: Frequency Distributions
frequency distribution: presents the score values and their frequency of occurrence. When presented in a table, the
score values are listed in rank order, with the lowest score value usually at the bottom of the table.
+ individual scores usually grouped into class intervals and presented as a frequency distribution of grouped scores.
When grouping data, important issue is how wide each interval should be
+ whenever data are grouped, some information is lost wider the interval, more information lost.
+ ie: large interval smooth display no zero frequencies ;; a lot of information is lost though. Scores distributed in the
interval are lost via grouping
- larger the interval, greater the ambiguity
narrower the interval, more original data are preserved
+ extreme case: interval reduced to one unit wide and back to the individual scores.
+ when interval is made too narrow, the values with zero frequency and an unclear display of the shape of the
distribution and its central tendency.
In grouping scores, there's a way of losing information & presenting a meaningful visual display
+ to have best of both worlds, choose an interval width neither too wide nor too narrow
+ determine interval width by dividing the distribution into 10-20 intervals
- within this range, specific number of intervals used depends on the number and range of raw scores
- more intervals used, the narrower each interval becomes.
Constructing a Frequency Distribution of Grouped Scores:
steps for constructing a frequency distribution of grouped scores:
+ 1. find the range of the scores
+ 2. determine the width of each class interval (i)
+ 3. list the limits of each class interval, placing the interval containing the lowest score value at the bottom
+ 4. Tally the raw scores into the appropriate class intervals
+ 5. Add the tallies for each interval to obtain the interval frequency
finding the range
+ range = highest score lowest score
+ ie: 99 46 = 53
determining interval width (i):
+ let's assume we wish to group the data into approximately 10 class intervals
+ i = range/number of class intervals
+ i =53/10
+ i =5.3 (round to 5)
+ when i has a decimal remainder, we'll follow the rule of rounding i to the sane number of decimal places as in the raw
scores. Thus, i rounds to 5.
listing the intervals:
+ we begin with the lowest interval
+ first step: determine the lowest limit of this interval
+ two requirements:
- 1. the lower limit of this interval must be such that the interval contains the lowest score
- 2. it is customary to make the lower limit of this interval evenly divisible by i.
+ the lower limit is assigned the value of the lowest score in the distribution if it's evenly divisible by i. If not, then the
lower limit is assigned the next lower value that is evenly divisible by i.
- ie: lower limit of the lowest interval begins with 45 because the lowest score 46 is not evenly divisible by 5.
+ once the lower limit of the lowest interval has been found, we can list all of the interval
- since each interval is 5 units wide, the lowest interval ranges from 45-49 although this interval is only 4 units wide, it'
really is 5 (45,46,47,48,49)
- in listing the other intervals, they must be continuous & mutually exclusive (intervals must be such that no score can be
legitimately included in more than one interval)
+ lit the apparent limits of each interval and omit listing the real limits
tallying the scores:
+ raw scores are tallied into the appropriate class intervals
+ tallying: a procedure whereby one systematically goes through the distribution and for each raw score enters a tally
mark next to the interval that contains the score
+ ie: 95 is placed in the interval 95-99 summing into frequencies:
+ tally marks converted into frequencies by adding the tallies within each interval
Relative Frequency, Cumulative Frequency, and Cumulative Percentage Distributions:
often desirable to express the data from a frequency distribution as a relative frequency, a cumulative frequency, or a
cumulative percentage distribution.
Relative frequency distribution: indicates the proportion of the total number of scores that occur in each interval
cumulative frequency distribution: indicates the number of scores that fall below the upper real limit of each interval
cumulative percentage distribution: indicates the percentage of scores that fall below the upper real limit of eachinterval
to convert a frequency distribution into a relative frequency distribution, the frequency for each interval is divided by the
total number of scores
+ Relative f = f/N
+ ie: relative frequency for the interval is 45-49 found by dividing its frequency (1) by the total number of scores (70)
relative frequency for this interval = 1/70 = 0.01
- relative frequency is useful because it tells us the proportion of scores contained in the interval
cumulative frequency for each interval found by adding the frequency of that interval to the frequencies of all the class
intervals below it cumulative frequency for the interval 60-64 = 4 + 4 + 2+ 1 = 1
cumulative percentage for each interval is found by converting cumulative frequencies to cumulative percentages
+ cumulative % = cumulative f/N x 100
+ for the interval 60-64
- cumulative % = cumulative f/N x 100
- cumulative % = 11/70 x 100
- cumulative % = 15.71%
cumulative frequency and cumulative percentage distributions are especially useful for finding percentiles and percentile
measures for relative standing
+ used extensively in education to compare the performance of an individual to that of a reference group
+ the 60 thpercentile point is the value on the measurement scale below which 50% of the scores in the distribution fall.
percentile/percentile point: the value on the measurement scale below which a specified percentage of the scores in
the distribution fall.