A better graphical tool to measure the “area under the curve” is the

Cumulative Density Function (CDF). A CDF plots the X categories

against the “probability of a value taking a value below the chosen X

value.”

The CDF for the Normal Density Function is reproduced in the next

figure. The curve increases from left to right (from 0 to 18). The height at

any X-value tells us “the probability of a value having a value below this

X-value equals the Y-axis value of the CDF at this X.”

The area under any Density Function curve always equals 1. The relative

8

frequency equals (frequency that X takes on this particular value) divided by (the

total sample size). Therefore, in a sense, the height gives the frequency weight for

each X value. If you sum all the relative frequencies, their sum is “sample size

divided by sample size” equals 1. This is the area under the curve. It can also be

expressed in percentage terms; the total percentage area then becomes 100%.

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Chapter 7: Probabiity Density Functions & Confidence Intervals

Figure 102: The “Cumulative Density Function” (CDF) associated with the Probability

Density Function (PDF) shown in the previous figure

The CDF is a better tool for answering the typical questions about the

properties of a data series. CDF is of great importance for building

Confidence Intervals and implementing hypothesis tests.

In fact, for some Density Functions, Excel only measures the CDF only

(and not the CDF & PDF).

The CDF and Confidence Intervals

The concept of a Confidence Interval for a measured parameter (typically

for a mean) is based on the concept of probability depicted by a Density

Function curve. A Confidence Interval of 95% is a range of X values

within whose range the sum of the relative frequencies is 0.95 or 95%.

I will use this property to show how to create Confidence Intervals for

various distributions using the Inverse of the CDF. (You will learn more

on the Inverse in the next sub-section.)

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Statistical Analysis with Excel

Table 14: Probability Density Function (PDF) and Cumulative Density Function (CDF)

Cumulative Density Function (CDF) &

Is there an option Probability Density Function (PDF):

to request the Information requirements for

Function Cumulative parameterization

Density Function

(CDF)? Std Degrees of

Mean Other

Dev freedom

TDIST Tails #

LOGNORMDIST

2nd degree of

FDIST

freedom

alpha, beta, upper and

BETADIST

lower bound

CHIDIST

NORMDIST

NORMSDIST

Alpha and

WEIBULL

beta

# of

NEGBINOMDIST (Probability)

successes

BINOMDIST (Probability)

EXPONDIST Lambda

Alpha and

GAMMADIST

beta

Sample size, population size, # of

HYPGEOMDIST

successes in population

POISSON

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Chapter 7: Probabiity Density Functions & Confidence Intervals

7.1.C INVERSE MAPPING FUNCTIONS

The Cumulative Density Function (CDF) tells us “For any X series, the

probability of the value of X falling below a specific x value can be

calculated from the height of the Cumulative Density Function (CDF)

at that x value.”

An inverse function does the reverse mapping: “For a probability P, the X

to who™s left the probability of the data lying can be obtained by a reverse

reading of the Cumulative Density Function (CDF). That is, from

“Desired Cumulative Probability unknown X that will give this desired

cumulative probability P.”

Alternatively, “Inverse” functions find the X value that corresponds to a

certain “probability of values below the X equaling a known cumulative

probability.”