Overview: An overview of the basic quantitative methods human resource personnel rely upon to administer compensation and benefits programs.
Most human resource practitioners realize in an intuitive way that they are using a mathematical system when they apply quantitative methods, but probably most would not be able to say what the actual system is. A mathematical system depends on:
While any system can be used, and human resource practitioners are accustomed to the simple rules and operations of the arithmetic system of real numbers, statistical practices often require the application of special rules and operations within the set of rules of the mathematical system. In a sense, that makes the statistical methods, a "subset" of the entire mathematical system that we use. This is essential to remember since statistical methods cannot exceed the limitations of normal mathematical boundaries. It is also important to remember that not all the tests or approaches may be appropriate, because the boundaries of normal mathematics might be exceeded in given situations.
In statistics, these boundaries are called assumptions.1
What are the assumptions that a normal statistical review (e.g., bell-shaped curve) might require in comparing the results of performance appraisals between men and women in your organization?
Mathematical Systems Solution
Without more detailed information, the example of appraised measurement exceeds all these boundaries.2
There are four basic kinds of measurements:
The first of these, nominal measurements (or scales) are composed of numerical values that serve to identify discrete categories; i.e., the numbers are labels for the categories and imply no quantitative (measurable) differences that can be handled meaningfully in numerical operations. The numbers on front doors, for instance, constitute such a scale; Social Security numbers are another example.
In salary surveys, the numbering of positions or companies is just such a scale; the numbers are used merely as labels.
|Company Name||Company Number||Position||Position Number||Salary*|
You are about to conduct a survey of five positions, and you have numbered them as follows:
What significant computations can be completed using the position numbers?
No SIGNIFICANT computations can be completed.
For example: 1 + 2 + 3 + 4 + 5 = 15 means nothing.
Nominal measurements allow only the most limited application. For example, one can count classes (the numbers of positions numbered "2" for instance). Or "Yes" - "No" can be counted where 0 - 1 correspond to "yes" and "no." However, most human resource administrators will find that using nominal measurements in any kind of mathematical operation is bound to lead to error. For instance:
|Employee's Social Security Number:||536-41-3760|
|Spouse's Social Security Number:||437-28-5621|
is a totally meaningless number.
Ordinal measurements have, in addition to nominal properties, rank differences. That is, the numerical value of these scales indicates that there are not only differences between categories but that these are quantifiable differences. An ordinal scale ranks "observations" with regard to the extent to which they possess more or less of a given quality. The ranks do not, however, indicate the degree of difference (how much more or less) of the property each observation has.
There are events with dimensions that cannot be readily quantified. It would be absurd to state that a painting is twice as beautiful as another or that one restaurant has food one-third as tasty as another. For this reason, ordinal scales are often applied to observed events that differ along qualitative rather than quantitative dimensions (especially when the qualitative dimension cannot be easily expressed quantitatively).
For example, in job evaluation plans, the following factors cannot be easily broken down into quantifiable units.
Can one job have two times as much the problem complexity as another? But within each factor there can be a breakdown of steps.
For example, problem complexity might be broken down by:
These steps may be described as "ordinally ranked."
You have been asked to estimate the average salaries of individuals within the United States who hold the following degrees:
Average salaries by degree:
|High School Diploma||$31,112|
This measurement is very important. Performance appraisals are almost always on ordinal scales, yet human resource practitioners often try to utilize mathematical tests and computations that require more stringent assumptions.
Interval scales are equal unit scales; that is, the distance between adjacent units on an interval scale is the same, irrespective of the magnitude of the adjacent scale units. For instance, the distance between $5,000 and $6,000 is the same as the distance between $61,000 and $62,000, since it reflects a measurable increase in real dollars.
Interval scales, however, are only relative. For example, if these amounts ($5,000 - $6,000 and $61,000 - $62,000) were to represent sales compensation, the amount of actual sales required to produce the increase in sales compensation in the above example may be different at $5,000 (to $6,000) and $61,000 (to $62,000). Thus, the dollar scale may reflect relative rather than absolute changes. For example:
Interval scales are the most frequently used measurement in human resource pay practices. However, many human resource measurements cannot satisfy the requirements that measurements be similar and equidistant in their measurement intervals. Their ordinal or nominal characteristics must be recognized in order to understand the explanations of tests discussed later in this chapter.
A salesman sells a product for $90,000. His employing broker's commission is 5% after the first $30,000. The salesman receives 60% of this broker's commission. What is the salesman's commission?
Revenue - Base = Commissionable Revenue
$90,000 - $30,000 = $60,000
Commissionable Revenue x Broker's Percentage = Total Commission
$60,000 x 0.05 = $3,000
Total Commission x Salesman's Percentage = Salesman's Commission
$3,000 x 0.60 = $1,800
Special note: These plans are often increased or decreased in specific percentage amounts for:
Salesmen operate (and are motivated) on ordinal rates via interval increases.
The ratio scale has, in addition to the properties of the nominal, ordinal and interval scales, an absolute zero. The absolute zero represents a point below which no value can be assigned. Zero represents no less than one of the property being gauged by the scale. In salary administration, for example, dollars in value can be shown on the interval scale, while percentages can be shown on a ratio scale. Compensation dollars, however, can be ratio ( i.e., if no negative exists).
(One must remember that division by zero is not an acceptable mathematical operation. Hence, ratio measurements have certain computational limitations.)
Examples of ratio measurements include:
|1,000,000||Rent = 5,000||0.5%|
|Travel = 50,000||5.0%|
|Salaries = 500,000||50.0%|
|Benefits = 175,000||17.5%|
You have three departments, each of which has budgeted the following salary increases for the coming year:
Department A: 7% raise for 2 employees
Department B: 9% raise for 3 employees
Department C: 11% raise for 95 employees
All employees are receiving $10,000 annual base salaries.
What is the overall budgeted salary increase for the coming year?
All individuals receive $10,000/year salaries.
|Dept.||Number of Employees||% Increase||Total Base
Overall increase of salary budget = $108,600 / (100 x $10,000)
Overall increase of salary budget = 10.86%
Overall increase of salary budget = 27 + 3
Overall increase of salary budget = 9%
This example is designed to illustrate a common error in human resource administration: addition and division of ratios. "Averaging averages" rarely works. As shown above, 9% is not the overall percentage of salary increase because of the equal weight given to departments rather than individuals.
The real average is determined by adding the values of the variables (increases) for all 100 observed events and then dividing that correctly by the number of observations (the total base employees' salaries).
The basic "statements" of operations in mathematics are statements of equality or inequality. The operation performed on a number or group of numbers is designed to reduce the value to a single number (essentially, to simplify).
The four most common mathematical operations are:
The process that combines several numbers to obtain a single number whose value equals the total value of the numbers.
This operation is usually indicated by the "+" sign.
The inverse (the process that reverses an original process) of addition: this operation is the reduction of a number's value by the value of another number (or several numbers in sequence).
This operation is usually indicated by a " - " sign.
The process that adds a number to itself a designated number of times; that is, 3 x 4 is merely four 3s added together or 3 + 3 + 3 + 3.
This operation is indicated by an "x" sign or sometimes by the centered period, " ." sign. It may also be understood when a symbol is juxtaposed to a number or another symbol (as in the operation fy, where the value is f x y. Multiplication is also shown if the symbol is next to an operation within parentheses, as in F(3 x 5) where the value is 3 times 5 times F. One always performs operations within parentheses first.
This operation is the inverse of multiplication; that is, the value of the number is reduced into equal parts by a designated number.
This operation is indicated by a "÷" sign or by the slanted or horizontal line in a ratio or fraction, as indicated by "/" in the fraction 3/4.
Remember: The operations of multiplication and division are performed first in most statistical packages, unless parentheses indicate a sub-operation to be performed first.
ADDITION: the process that combines values
|a/b + c/d||=||
ad + cb
|3/7 + 2/9||=||
(3 x 9) + (2 x 7)
(7 x 9)
27 + 14
SUBTRACTION: the process that reduces values
|a/b - c/d||=||
ad - cb
|3/7 - 2/9||=||
(3 x 9) - (2 x 7)
(27 x 9)
7 - 14
MULTIPLICATION: the process that increases the given number
|a/b x c/d||=||
a x c
b x d
|3/7 x 2/9||=||
3 x 2
7 x 9
DIVISION: the process that reduces the value equally
|a/b ÷ c/d||=||
a/b x d/c
a x d
b x c
|3/7 ÷ 2/9||=||
3 x 9
7 x 2
The simplest mathematical computation is addition. Addition combines numbers to make a total called the sum. Of all computations, however, addition, together with its inverse, subtraction, is the source of most computation errors. For years, the IRS has listed addition and subtraction mistakes as the leading cause of faulty returns.
For 2002 and beyond, companies must adopt a vesting schedule at least as generous as those below for employee 401(k) plans:
Six-year graded vesting proceeds like this:
|Years of Service||Unforfeitable Percentage|
You are 40 years old and have worked for 5 years for your firm. As a participant in the firm's qualified pension plan, what might your non-forfeitable vested pension percentage be in each of the above plans?
Under the 2 different systems, you would be vested as follows:
|3-year cliff vesting||100%|
|6-year graded vesting||
Subtraction is the inverse of addition. It calls for the difference in the values of numbers.
Integration with Social Security benefits is a perfect example of how human resource administrators utilize subtraction:
|Individual retirement benefit from a retirement plan||$600/month|
|Amount to be received from Social Security||$400/month|
|Employer's contribution percentage||37.5%|
|Employer's portion of $400 Social Security payment||$150/month|
|FINAL RETIREMENT BENEFIT FROM PLAN||$450/month|
Another example would be long-term disability integration and net vs. gross pay computations.
An employee who would receive a final pension of 50% of last year's earnings in New York decides to retire at age 60 rather than at age 62 (when he would be eligible for the full 50% benefit). What might his percentage be? Remember that an actuarial reduction for early retirement in a defined benefit plan is roughly 5% per year of the benefit (e.g., 3% of a 60% defined benefit).
5% of benefit per year = 50% x 5% = 2.5%
(i.e., .50 x .05 = .025 or 2.5%)
50% - 2.5% per year = 50% - (2.5 x 2)
= 50% - 5%
FINAL PENSION = 45%
Municipalities are often not part of the Social Security System, so integration does not apply. In some places, for example, retirement pay is based on a percentage of "last year's earnings"; this includes overtime, hence, occasions have arisen when retirement pay is almost at full last year's base pay.
Multiplication is the computation process that simplifies addition. Factors are combined to give a product.
The compounding of interest refers to the common savings account approach that permits the following increases:
|Year 1:||Interest is earned on the sum deposited. (e.g., at 4%, $10,000 would earn $400 in the first year.)|
|Year 2:||Interest is earned on the original sum deposited (i.e., $10,000) and on the previous year's interest (i.e., $400 more) if the interest is left in the same account. At 4%, $10,000 would still earn $400 in the second year, but now the $400 from the previous year's interest would also be earning 4% interest to make a total of $416 earned in the second year. This makes a total of $10,816.|
|Year 3:||This process continues indefinitely provided the accumulated interest is left in the same account. For example, the amount in the example above would begin the third year with $10,816 earning 4% on the entire amount.|
The formula reads:
Ending Sum = Initial Sum x (1 + Interest Rates)Years
e.g., $ 10,816 = $ 10,000 x (1 + .04)2 *
*It should be remembered that the exponent (power) means that the operation or number should be multiplied by itself that many times. In the example, therefore, the process is (1.0 + . 04)(1.0 +.04). Always perform the operation in parentheses first.
It is the Year 2000 and the average BA starting salary is $30,000. You predict these starting salaries will increase at a rate of 9% per year. Using the compound interest approach and a calculator, determine a starting salary for an MBA in the year 2016.
|Year||Principal ($)||Interest at 9% ($)|
Amount x (1 + Interest Rate)Years
= $30,000 x (1 + 0.09)(2016 - 2000)
= $30,000 x (1.09)16
= $30,000 x 3.970305
Division is the inverse of multiplication. It has the same relationship to multiplication that subtraction has to addition. An inverse cancels the process of its inverse. For example, if you begin with 3 and multiply it by 5 (3 x 5), you need only divide the result (15) by 5 to once again have three. In other words it reversed the original process.
In a division problem such as 15 ÷ 5 = 3, 15 is the dividend, 5 is called the divisor, and the result (3) is the quotient.
A very useful thing to remember in human resource operations is called "The Rule of 72." If you divide any interest rate into the number 72, your answer will be the number of years it will take for a compound interest rate to double the original sum.
It is the year 1984 and the average MBA starting salary is $30,000. You predict starting salaries to increase at a rate of 9% per year. Using the Rule of 72, what year will a starting salary for an MBA double to be $60,000?
Divide the percentage into 72.
Answer = the number of years required to double the principal.
72 ÷ 9 = 8
7.2% doubles money in ten years
72 ÷ 7.2 = 10
12% doubles money in six years
72 ÷ 12 = 6
If the inflation rate in 2000 is 12%, how much will a loaf of bread cost in the future?
The following sections will show you how to use these mathematical operations for human resource planning, including to determine:
Salary surveys deal with money, and money accrues a certain value over time. In the traditional sense, money can always be earning interest. The value of $1,000 today is not necessarily what the value of $1,000 might be next year if that sum can earn 10%.
Many human resource decisions involve deferred payments. The collection of their values in a survey would be misleading if they were reported only at their face values.
The computation of the value of the present worth of any sum to be paid in "n" years can be computed thus:
A retirement plan promises to pay an individual $10,000 in ten years. If present rates of interest are 10%, what will that $10,000 be truly worth to the individual today?
Present value of compensation dollars depends also on effective tax rates. It is important to remember that in the U.S., maximum tax rates were over 90% 50 years ago. Computations must all estimate this effect -- a difficult task at best.
A logarithm is the exponent of the power to which a fixed number must be raised to produce a given value. For example, if the fixed number is 10 (the most common base), the logarithm of 1,000 is 3. You have to multiply 10 times itself three times (10 x 10 x 10) to produce the desired result (1,000). The logarithm of 10,000 is four.
Logarithms are commonly used in human resource surveys, especially in graphs that show salaries versus some size dimension. The reason for their use is that they allow visual comparison of data that may be quite dissimilar in size.
Companies of the above sizes could all be shown on the same graph with divisions from 1 to 10.
Working with logarithms requires the exercise of manipulating the exponents (power of the numbers).
Logarithms at a base of 10 are best illustrated by:
Dividing or multiplying with logarithms is like dividing or multiplying with other exponents. To multiply, you add the exponents. To divide you subtract the exponents. For example:
103 x 104 = 107
108 ÷ 102 = 106
Compute the average sales of the four companies above by using the logs. (The log of four is 0.6021.)
If 10 were divided by 4, one would find a logarithm of:
log of 10 = 1
log of 4 = 0.6021
log of 10 ÷ log of 4 = 1 - 0.6021 = 0.3979
Looking up the value of 0.3979 in Table 2 one finds 2.5. With a "0" characteristic, one moves the decimal place from the left one digit -- that is, 2.5 (1.3979).
Average Sales Size is:
The log of which (counting digits and using Table 2)
(log of 4)
9.8139 = $6,515,000,000
As shown in the preceding example, the mean or average of a distribution is defined as the sum of the values of the variables in the distribution divided by the number of observations in that distribution. When the distribution is a sample:
AVERAGE = Sum of Variables / Number in Sample
Remember: An observation is a discrete event to which the value of a variable has been assigned; several observations can have the same value.
A variable is the aspect of the world being observed, to which value has been assigned.
The number of observations in the sample is 20. (Observations could be number of employees in different departments, number of days worked overtime, etc. -- whatever can be defined as being distinct and separate.) The sum of the variables (i.e., the values of each variable) is 529. (For instance, one department may have 40 employees, another 39, another 28; the 28 variables observed is the number of employees, the 20 observations are the departments.)
Average = 529/20
Average = 26.45
Mean = Average
Mean = 26.45
You conduct a survey of controllers in companies in a certain industry and find the following salaries being paid: $14,000, $15,000, $15,000, $16,000, $18,000, $21,000, $28,000, and $42,000. What is the average salary?
AVERAGE = Sum of Variables / Number in Sample
Sum of Variables = $14,000 + $15,000 + $15,000 + $16,000 + $18,000 + $21,000 + $28,000 + $42,000 = $169,000
Number in Sample = 8
Average = $169,000 / 8 = $21,125
The median is the midpoint of a distribution. Half of the observations in a distribution are above the median; the other half are below.
When the sample or population consists of an even number of observations, the true median may lie halfway between the two middle observations.
|Table A||Table B|
|THE MEDIAN IS 26||THE MEDIAN IS 24|
You conduct a survey of controllers' salaries in companies in a certain industry and find these: $14,000, $15,000, $15,000, $15,000, $16,000, $16,000, $18,000, $21,000, $28,000, S42,000. What is the median salary?
Calculation of the median:
A general rule in statistics is to round up on odd counts from the lower observation ($16,000); but because this is the real world, we would suggest utilizing the real median which is $17,000, a value at the halfway point which is at an equal distance from each variable.
However, some texts state that it must be an actual observation (in this case, an actual salary), which means that it would not have been $17,000.
A popular survey gives the following definition of "weighted average."
The average weekly salary reported by a company for a given position is multiplied by the number of employees in the job. The results are totaled for all companies reporting the position, and then divided by the total number of incumbents.
Another survey defines weighted averages a "the distribution of salaries is reviewed by deleting the top 25% and bottom 25% of the sample. The average is then computed from the interquartile range."
The point to note, like the rounding of medians, is that in salary surveys this term "weighted average" may have different meanings for different surveys. Most commonly, the first definition is used.
The reason for weighting averages is that the result represents the average of a total population and not just a subset. The equation for this procedure reads:
a, b, c are averages of measurements
N1, N2, N3 are numbers of measurements
Two companies reported the average salary for a similar position as follows:
|Company A||Company B|
What is the weighted average for this position? How does it compare to simply averaging the two surveys?
|Company A||Company B|
|Weighted Average||(10.00 x 6) + (8.50 x 2) = 77.00/8 = 9.63|
|Simple Average||10.00 + 8.50 = 18.50 = 9.25|
The mode is that category of the distribution that contains observations that appear with the greatest frequency. That is, the most frequent set of measurements is the mode of the distribution. In grouped data, the mode is associated with the midpoint of the category that has the greatest frequency.
The category with the greatest frequency concentration often tends to be located at or near the center of a distribution. However, this is not always the case; thus, as a measurement, the mode leaves much to be desired.
|39||The mode is 39.|
|39||This is the observation with the greatest frequency.|
You conduct a survey of controllers' salaries in a certain industry and find the following salaries:
What is the mode salary?
The highest number of occurrences of any single event -- in this case, 3 -- is $15,000.
Therefore, $15,000 is the mode.
Percentages are the most commonly used form of fractions. Computed in 100ths, they allow a representation of a fraction in terms of 100s or "cents" (from the Latin), providing the user with a basis for making comparisons.
Percentiles are arbitrarily selected units determined by dividing a whole into a distribution of 100 equal parts. A percentile distribution is based on the number of observations constituting a given percentage of the total number of observations in the distribution, irrespective of category; percentiles are frequently used to determine what proportion of the distribution falls below a given level.
You have started a company and have hired an administrative assistant at $28,000. He is the tenth highest paid of 123 secretaries. His salary is at what percentile?
The correct answer is "91st percentile."
|Rank||10th highest of 123|
|Standing||Maximum minus ranking
123 - 10 = 113
(Standing divided by maximum) x 100
(113 / 123) x 100
0.9187 x 100
|Percentile||91th percentile (In determining test scores, it is a common practice to drop the decimal place and round down.)|
The bottom line of a graph (the horizontal line) is known as the base line for as the "horizontal axis." By convention, mathematicians refer to it as the x-axis.
The line perpendicular to the x-axis (usually at the left) is known as the "vertical axis" or, again according to convention, the y-axis.
In equations, these axes are often defined in the form of:
value of y = value of x(another value) + a constant
That is: y = mx + b
You conduct a survey of typist test scores and find the following results: 100, 85, 70, 75, 50. Correspondingly, you find the age of each typist to be 60, 50, 40, 30, 20. What would be the x-axis if you were to plot this data?
Either salaries or years of experience could serve as the x-axis.
An equation is most often expressed in the form:
y = mx + b
The slope is the change in the value of y that corresponds directly to a change in the value of x.
An equation is useful because it describes any value of y if you know any value of x.
Draw a graph that shows the following three lines:
|The first line should show where the "x" value always equals the "y" value.||
y = 1x + 0
y = x
|The second line should show where the "y" value is 10 less than the "x" value and an additional 10% more than the "x" value.||
y = (x - 10) 0.10
y = 0.1x - 1
|The third line should show the "y" value that is constant regardless of the value of "x."||
y = 0x + constant
y = constant
The equation of a straight line is:
y = mx + b
When given any two points on a line, one can calculate an equation to find any other points on the same line as follows:
|Point 1 (x1y1)||Point 2 (x2y2)|
|Intercept (b)||y1 - mx1|
|Intercept (b)||y2 - mx2|
You have seen that a 60-year-old typist scored 100, while a 20-year-old typist scored 50 on a typing test. Calculate the equation for the line that passes through these points.
Point 1 (60,100)
Point 2 (20,50)
|Slope (m)||= 1.25|
= y1 - mx1
= 100 - (1.25 x 60)
= 100 - 75
|Intercept (b)||= 25|
The equation for the line that passes through (60,100) and (20,50) is:
y = 1.25x + 25
To check, review y2 and x2.
y2 = mx2 + b
50 = (1.25) (20) + 25
50 = 50
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Internet Based Benefits & Compensation Administration
Thomas J. Atchison
David W. Belcher
David J. Thomsen
ERI Economic Research Institute
Copyright © 2000 - 2013
Library of Congress Cataloging-in-Publication Data
HF5549.5.C67B45 1987 658.3'2 86-25494 ISBN 0-13-154790-9
Previously published under the title of Wage and Salary Administration.
The framework for this text was originally copyrighted in 1987, 1974, 1962, and 1955 by Prentice-Hall, Inc. All rights were acquired by ERI in 2000 via reverted rights from the Belcher Scholarship Foundation and Thomas Atchison.
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