REAL LIMITS
Part 1: Getting the Basic Concept
of Real Limits
Source: http://www.uop.edu/cop/psychology/Statistics/sigfig.html
Real Limits- When measuring a continuous variable like time, for example, we must understand that the
measure is only an approximation. If we say that a sprinter ran the one hundred meters in 9.4 seconds,
we realize that the measurement could range from 9.35 to 9.45 seconds. That is, the runner's time was
between 9.35 and 9.45 seconds or 9.4 +/- .05 seconds. These numbers are called the real limits of that
measure.
Source: http://www.uwsp.edu/psych/stat/2/prelim.htm#RealLim]

Since continuous numbers are rounded, they are only approximate. Thus, the number 33 more precisely
falls in the range of 32.5-33.5.

More generally, the real limits of a number equals the number plus and minus () 1/2 the unit of
measurement.

Examples:
Number
Real Limits
Unit of 1/2 Unit of
meas. meas.
Lower Upper
33.3
.1
.05
33.25
33.33
.01
.005
33.325 33.335
33.333
.001
.0005
33.3325 33.3335
1
.5
33
32.5
33.35
33.5
Source: http://visualstats.org/vista-frames/help/class-notes.html
Histograms and Real Limits
A histogram is used to portray the (grouped) frequency distribution of a variable at the interval or ratio
level of measurement. It consists of vertical bars drawn above scores (or score intevals) so that
1. The height of the bar corresponds to the frequency
2. The width of the bar extends to the real limits of the score (interval)
Real Limits, Apparent Limits, and Frequency Distributions
o
Recall that a continuous variable has an infinite number of possible values. It can be represented by a
number line that is continuous and has an infinite number of points. However, when we measure a
continuous variable we have only a finite measurement process, resulting in numbers that have a finite
precision.
o
If our measurements of a continuous variable are all numbers that are all in whole integer units, our
precision of measurement is 1 unit. In this measurement situation, an observed value of 8 would be
obtained when the "real value" is 7.8 or 8.21, or any other value between 7.5 and 8.5. Thus, the
observed value of 8 actually represents a range of "real values" from 7.5 to 8.5. These values are called
the "real limits".
o
The concept of "real limits" also applies to class intervals. In the table at the
right the interval denoted as "60-64" actually has "real limits" of 59.5-64.5. The
values denoting the interval as 60-64 are called the "apparent limits".
Part 2: Real Limits in Chapter 6
(Normal Distribution, Probability)
Source: http://www.psychstat.missouristate.edu/introbook/sbk10m.htm
Probability = Area Under the Curve Between any Two Scores
Although probability is a common term in the natural language, meaning likelihood or chance of
occurrence, statisticians define it much more precisely. The probability of an event is the theoretical relative
frequency of the event in a model of the population.
The models that have been discussed up to this point assume continuous measurement. That is, every score
on the continuum of scores is possible, or there are an infinite number of scores. In this case, no single score
can have a relative frequency because if it did, the total area would necessarily be greater than one. For that
reason probability is defined over a range of scores rather than a single score. Thus a shoe size of 8.00
would not have a specific probability associated with it, although the interval of shoe sizes between 7.75 and
8.25 would.
An Example: Shoe Sizes
Let’s say we want a graph of the distribution of
women’s shoe sizes. In actuality shoe size a
discrete variable, on an interval scale, so we
would use a histogram. We can also use a
frequency polygon, and smooth out the curve,
so that it would look like a normal curve.
Suppose that the frequency polygon of shoe
size for women actually looked like the
following:
If this were the case the proportion (.12) or percentage (12%) of size eight shoes could be computed by
finding the relative area between the real limits for a size eight shoe (7.75 to 8.25). The relative area is
called probability.
The probability model attempts to capture the essential structure of the real world by asking what the world
might look like if an infinite number of scores were obtained and each score was measured infinitely
precisely. Nothing in the real world is exactly distributed as a probability model. However, a probability
model often describes the world well enough to be useful in making decisions.
Source: http://visualstats.org/vista-frames/help/class-notes.html
Percentiles & Percentile Ranks
In order to define what we mean by percentiles and percentile ranks, we first begin by defining cumulative
frequencies and cumulative percentages.
Cumulative Frequencies:
The cumulative frequency of a particular category in a frequency table or distribution is the number of
observations in or below that category.
Cumulative Percentages:
The cumulative percentage of a particular category in a frequency table or distribution is the percentage of
observations in or below that category.
In the following table, the "cf" column presents cumulative frequencies and the "c%" column presents
cumulative percentages.
_________________
X
f
cf
c%
5
1
20 100%
4
5
19
95%
3
8
14
70%
2
4
6
30%
1
2
2
10%
A "cf" value of a category is the sum of the frequencies in the categories at and below the category in
question.
A "c%" value of a category is 100 times the cumulative frequency of the category, divided by the total
frequency. That is
c%=100*(cf/N)
Percentile Rank
The percentile rank of a particular score is defined as the percentage of the scores in a distribution which are
at or below the particular score.
It is possible to determine some percentile ranks directly from the frequency distribution table, provided that
the percentile ranks are percentages that appear in the table.
For example, the percentile rank of X=3.5 is 70% (note that 3.5 is the upper real limit of the category where
X=3).
Percentile
When a score is identified by its percentile rank, the score is called a percentile.
It is also possible to determine some percentiles directly from the frequency distribution table, provided that
the percentiles are upper real limits of a category.
Thus, the 95th percentile is X=4.5, the upper real limit of the X=4 category.
Interpolation
The process known as interpolation must be used to estimate percentile ranks and percentile for values not
given in the table.
The interpolation process is explained in the textbook on pages 54-57.
A note on The Normal Distribution
In the normal distribution, frequencies, actually proportions, correspond to an area.