Figure 103: Reading inverse mapping from a Cumulative Density Function (CDF). The

arrows show the values below which are 95% of the values of the data series.

Inverse functions permit easy construction of Confidence intervals.

This will be shown several times in further sections whenever I discuss

the construction of Confidence intervals.

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Statistical Analysis with Excel

Table 15: Inverse functions (also used to create Confidence intervals). Samples will be

available at http://www.vjbooks.net/excel/samples.htm.

Information

required by all Other information

Inverse Function

requirements

inverse

(“probability to

functions

Function value”) of this

Probability for

Cumulative Density Degrees

which the Std

Function (CDF)? of

Mean Other

Dev

corresponding

freedom

value is sought

TINV TDIST

LOGINV LOGNORMDIST

Second

FINV FDIST degree of

freedom

alpha,

beta,

upper

BETAINV BETADIST

and

lower

bound

CHIINV CHIDIST

NORMINV NORMDIST

NORMSINV NORMSDIST

NORMAL DENSITY FUNCTION

7.2

The Normal Density Function has several properties that make it easy to

make generalized inferences for the attributes of a series whose Density

Function can be said to be “Normal.”

Figure 104: A Normal Probability Density Function

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Chapter 7: Probabiity Density Functions & Confidence Intervals

Symmetry

The major measures of central tendency ” the mean, median, and mode

” all lie at the same point right at the place where the bell shaped curve

is at its greatest height.

The Density Function is perfectly symmetrical around this “confluence” of

central tendencies. Therefore, the left half of the Density Function

(measured as all points to the left of the mode/median/mean) is a mirror

image of the right half of the Density Function.

This is shown in the next figure ” the lighter shaded half is a mirror

image of the darker shaded half. So, the frequency of the values of the

variables becomes lower (that is, the height of the curve lowers) as you

move away from the mode/mean/median towards either extreme. This

change is gradual and occurs at the same rate for negative and positive

deviations from the mean.

Figure 105: An idealized “symmetrical” Normal Density Function. Note that the relatively

lightly shaded half is a mirror image of the relatively darker shaded half

The symmetry also implies that:

(a) The Density Function is not “skewed” to the left or right of the

mode/median/mean (and, thus, the Skewness measure = 0)

(b) The Density Function is not “too” peaked (which would imply that the

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Statistical Analysis with Excel

change in probability is very rapid when moving from the

mode/median/mean towards an extreme) nor “too” flat (which

would imply that the change in probability is very slow when

moving from the mode/median/mean towards an extreme).

The first property implies that Skewness = 0, and the second

implies that Kurtosis = 0.

Convenience of using the Normal Density Function

If a series is Normally distributed, then you just need two parameters for

defining the Density Function for any series X” the mean and standard

deviation of the variables values! This is because, once you know the

mean, you also know the mode and median (as these two statistics equal

the mean for a Normal Density Function).

Once you know the standard deviation, you know the spread of values

around the mean/mode/median. (A series that follows a Normal Density

Function is not skewed to the left or right, nor is “too” peaked or “too”

flat.)