"Describing Data with Number Summaries" 
Index to Module One Notes 
Measures of Central Tendency
In describing a set of data, we are
interested in the numerical measures of center and spread. Recall in
the cycle time example of Module 1.2, the center of the data appears
to be a cycle time of 21 days. That is formally called the measure
of central tendency of the data. There are two classic ways to
measure the center of a distribution of data: the mean and the
median.
Mean
The mean or average is the arithmetic measure of central
tendency, and is simply the sum of all of the observations in a
set of data divided by the total number of observations. So, the mean
of seven cycle times with values 21, 23, 19, 22, 20, 22, 20
is:
Mean = (21 + 23 + 19 + 22 + 20 + 22 + 20) / 7 = 21
You will read in statistics texts that this is
the formula for both the population mean (its symbol is the Greek
letter mu) and the sample mean (its symbol is an x with a bar over
it, or xbar). Don't get alarmed  this is not a course with a lot of
formulas, symbols and computations  we will let the computer do it.
I just want to highlight the computations for the classic measures in
descriptive statistics. You will also read in textbooks that measures
based on a population are called parameters, and measures
based on a sample are called statistics.
Median
The median is called the location measure of central
tendency since it is the middle value of an ordered array of
observations  50% of the observations in a set of data fall below
the median provided the data is in an ordered array. Since we have to
account for 100% of the data, 50% of the observations fall above the
median. So, given the cycle times of 21, 23, 19, 22, 20, 22, 20;
first created an ordered array (sorted list of the
numbers):
Ordered array = 19, 20, 20 , 21, 22, 22, 23
The median is the ordered or ranked observation that appears at location = (n + 1)/2, where n is the symbol for the number of numbers in the set of data. For this ordered array:
Median = (7 + 1) / 2 = 4th ordered observation, which is the number 21.
Do you note that the mean and the median are equal? They will be approximately equal (such as within 5% of each other) as long as the distribution of the set of data is somewhat symmetric without extreme values (signals or outliers) to one side or the other. This is an important point. Suppose we had an outlier or signal cycle time of 38 instead of 23. Let's calculate the mean and the median:
Mean = (19, 20, 20, 21, 22, 22, 38) / 7 = 23.14Median = (7 + 1) / 2 = 4th ordered observation, which is 21
The mean was "pulled" to the outlier value of
38  the mean is not resistant to extreme values, but the median is
since it is a location and not arithmetic measure. Note that the mean
is about 10% greater than the median: 21 is more representative of
the center when the distribution is not symmetric. As an aside,
economists generally use the median when reporting the measure of
center for housing values since sets of housing data are often skewed
by high or low extreme values. It would be misleading or even
unethical to report the mean housing value as a measure of center
when there are extreme values on the high side  the median is more
representative of center in that case. Also note that when the mean
> median, the distribution is skewed right  you saw a
histogram picture of that in Worksheet 1.2.3 of Module 1.2.
Similarly, when the mean < median, the distribution would be
skewed left  you saw a histogram picture of that in Worksheet
1.2.4 of Module 1.2.
Before we go on, let me show you how to compute the median with an
even set of data. Suppose we have the numbers 19, 20, 20, 21, 22, 22,
23, and 23. The median is:
Median = (n + 1) / 2 = (8 + 1) / 2 = 4.5th ordered observation; the number at the 4.5th location is the average between the 4th and 5th numbers or 21.5
Mode
Sometimes you hear of the mode as a measure of central
tendency. However, the mode is simply the most frequently appearing
number in a data set, so there are actually two modes in the above
distribution, 20 and 22. We will not be using the mode as a measure
of central tendency, but only to help in general examination of
shape, as discussed in Module 1.2.
Pause and ReflectThe mean and the median are measures of center. The mean and median will be approximately equal for symmetric sets of data. The median is resistant to extreme values, whereas the mean is pulled towards them. When the mean > median, the distribution is skewed right by an extreme value(s) > mean. When the mean < median, the distribution is skewed left by an extreme value(s) < mean. Thus, for sets of data with extreme values, the median is more representative of the center, and is the preferred measure. Alternatively, the extreme values could be removed from the data for separate study, and a new, more accurate arithmetic mean can then be computed.
Measures of Variation
Range
Equally important to describing the center of a set of data with
a numerical measure, is describing the spread or variation in a set
of data. There are three classic ways to measure variation. The
simplest measure of spread is the range. In the data set, 19,
20, 20, 21, 22, 22, 23:
Range = Maximum Number  Minimum Number = 23  19 = 4
So there is a spread of four days between the
smallest and largest cycle times.
Standard Deviation
The range is not very rich  it ignores all but two numbers. If
the distribution is approximately normal or a symmetric bellshape,
the classic measure of variation is the standard deviation. To
set the stage for the standard deviation, recall that the mean of
these numbers is 21. The number 19 varies from 21 by  2, the number
20 varies by 1, the number 23 by +2, and so forth. The standard
deviation is simply a measure of the average of these deviations. To
compute the standard deviation for a population of numbers 19, 20,
20, 21, 22, 22, 23; first find the mean (we already did that, it is
21). Now subtract each number from the mean, square the result (to
get rid of plusses and minuses), sum the resulting squared
deviations, and divide by the number of numbers. This is called the
variance of the population. The standard deviation of the population
(the symbol is the Greek letter sigma) is the square root of the
variance:
Mean = (19 + 20 + 20 + 21 + 22 + 22 + 23) / 7 = 21Sum of Squared Deviations = (19  21)^{2} + (20  21)^{2} + (20  21)^{2} + (21  21)^{2} + (22  21)^{2} + (22  21)^{2} + (23  21)^{2} = 4+1+1+0+1+1+4 = 12
Variance = 12 / 7 = 1.714
Standard Deviation = Square Root of 1.714 = 1.3
The standard deviation for a sample (the symbol is the letter s) is similar to the above, but the variance is computed by dividing the sum of squared deviations by the number of numbers minus one (n  1). In Modules 1.4 and 1.5 we will adjust the sample size to n  1 again, and refer to this as degrees of freedom. For large sample sizes, the impact of dividing by n  1 versus dividing by n is negligible. For small samples, it makes a difference so dividing by n  1 gives a larger or more conservative estimate of the spread. If the above set of data is a sample, the variance and standard deviation are:
Mean = 21
Sum of Squared Deviations = 12
Variance = 12 /(7  1) = 2
Standard Deviation = S = Square Root of 2 = 1.4
The standard deviation is interpreted with respect to the mean. For symmetric bellshaped sets of data, the interpretation of the standard deviation is:
95% (most) of the observations fall between the mean + 2 Std Dev
This can be combined with areas incorporating 1 and 3 standard deviations to completely describe a set of approximately symmetric, bellshaped data:
68% of the observations fall between the mean + 1 Std Dev95% of the observations fall between the mean + 2 Std Dev
99.7% (almost all or 100%) of the observations fall between the mean + 3 Std Dev
This very important property is called the
Empirical Rule  it applies to all symmetric bell shaped sets
of data. We will apply it later when we get to a larger example.
Interquartile Range
The third measure of spread or variation is the interquartile
range (IQR). To compute this measure, we need the 1st and 3rd
quartiles (Q1 and Q3, respectively). Twenty five percent of the
ordered observations in a set of data is contained within a quartile,
so 25% of the observations are below the first quartile, 50% of the
observations are below the 2nd quartile (Q2, also called the Median),
and 75% of the observations are below the 3rd. The IQR is simple Q3 
Q1. Here are the computations, using the numbers 19, 20, 20, 21, 22,
22, 24.
Q1 = (n + 1)/4 ordered observation = 0.25 * (7 + 1) = 2nd ordered observation which is the first number 20.where "*" is symbol for multiplicationQ3 = 3*(n + 1)/4 ordered observation = 0.75 * (7 + 1) = 6th ordered observation which is the second number 22 in the ordered array.
IQR = Q3  Q1 = 22  20 = 2 days
The middle 50% of the data falls within the IQR. Sometimes, analysts like to expand the location measure of spread by reporting the five number summary:
Minimum, Q1, Median, Q3, Maximum
Pause and ReflectThe standard deviation and the interquartile range are measures of spread or variation. The standard deviation is associated with the mean and is an accurate measure of spread for symmetric bellshaped distributions. The IQR is generally associated with the median, and is an accurate measure of spread for nonsymmetric distributions. With two numerical measures (mean and standard deviation; or median and IQR) we gain much knowledge about a data set  even data sets of thousands of numbers.
Relative Measures
Symmetry
There are some quick numerical measures than can be used to
supplement frequency distribution charts to determine if the
distribution is symmetric, bellshaped or skewed. They
are:
Set of Data is skewed to left if mean < median
Set of Data is skewed to right if mean > median
Set of Data is symmetric, bell shaped if:Mean is approximately equal to median
Empirical rule is met
IQR is approximately equal to 1.33 * Standard Deviation
We will try the above rules of thumb when we
illustrate the Excel descriptive statistics with a larger sample
later in this module.
Variation
The coefficient of variation is a useful tool when comparing the
variation of one distribution to that of another, especially if the
distributions have different units of measurement.
Coefficient of variation = (standard deviation / mean) * 100%
For the example: (1.4 / 21) * 100% = 6.67%
Suppose another company experiences mean cycle
time of 21 in their supply chain as well, but the standard deviation
is 7 days. The coefficient of variation in this case is 33%, which
reflects a process with much greater variability than one with 6.67%
coefficient of variation.
Observations
Relative measures of observations are useful to give observations
context within a data set. The percentile is a relative
measure of location for a number in an ordered array. We already
discussed examples of percentiles, since the first quartile is
actually the 25th percentile  25% of the data in an ordered array
are below the 25th percentile. Perhaps you recall your percentile
score on college entrance exams. If you scored in the 90th percentile
on the SAT, 90 percent of the ordered scores were below yours. To
find the 90th percentile, order the observations, then compute 90th
percentile = 0.90*(n+1) to find the ordered observation of interest.
If n = 780, then we would be looking for the 703rd number.
For distributions that are approximately symmetric, bellshaped, the
ZScore or Z Value is a powerful relative measure for
observations. The ZScore is simply the number of standard deviations
an observation is from the mean. So when we find the Z Score of a
number, we standardize that number. What is the ZScore for
the number 19.6 in a distribution with a mean of 21 and standard
deviation of 1.4?
Z = (observation  mean)/standard deviation = (19.6  21)/1.4 = 1
We would say that the number 19.6 is 1 standard
deviation from (to the left of) the mean of 21.
ZScores have associated probabilities. Do you recall that 68% of the
observations in a symmetric bellshaped distribution fall between the
mean and + 1 standard deviation? That "1" standard deviation
is the ZScore! Note also the 68%  we can find the probability for
getting any ZScore as long as the set of data is normal or is
approximately symmetric bellshaped. For example, what is the
probability of getting an observation less than 19.6 days? We know
that theoretically, the range of data from 19.6 to 22.4 days includes
68% of the data, and since we have to have 100% of the data accounted
for, 100%  68% or 32% of the data must include numbers up to 19.6
and above 22.4. Since we assume these distributions are symmetric,
16% of the data is up to 19.6 and 16% is above 24. So, for a
symmetric bell shaped distribution with mean of 21 and standard
deviation of 1.4, the probability of observing a number less than
19.6 is 0.16 or 16%.; the probability of observing a number greater
than 19.6 is simply 100%  16% or 84%.
Using Excel for the
Computations
We will use the Data Analysis
Tool and Statistical Functions to produce the descriptive
statistics we have discussed thus far in Module 1.3.
The quickest way to generate a family of descriptive statistics is to
use the Data Analysis Descriptive Statistics Tool. Lets use
the data from Worksheet 1.2.1 in Module 1.2. First create a column of
the cycle times as shown in that Worksheet (or if you saved that
example, copy C1:C31 to a new area of your worksheet past the
histogram, such as Column J). I placed the numbers in column J,
starting with the title "Time" in Row 1, and the 30 cycle times in J2
to J31. The data doesn't have to be sorted to compute the descriptive
statistics, but it can be. Now select Tools from the Standard
Toolbar, then Data Analysis from the pulldown menu, then
select Descriptive Statistics and follow the dialog box
requests as you did in creating the histogram. For output options,
select Summary Statistics and Confidence Level for the
Mean, leaving the confidence level default at 95%. I also like to
put the output close to my data and histogram if I created a
histogram for the data. Remember that you only have to enter one cell
location for the Output Range, such as L1, and that cell will
define the upper lefthand corner of the range of cells needed for
the output  just be sure that the worksheet is clear below and to
the right of the cell you select. Note that you can put your output
on a separate worksheet or even in a separate workbook if you like.
You should get Worksheet 1.3.1 below:
Worksheet 1.3.1
Time Mean 21.07 Standard Error 0.54 Median 21 Mode 21 Standard Deviation 2.94 Sample Variance 8.62 Kurtosis 0.90 Skewness 0.89 Range 13 Minimum 16 Maximum 29 Sum 632 Count 30 Confidence Level(95.0%) 1.10
We have discussed the Mean,
Median, Mode, Standard Deviation, Sample Variance,
Range, Minimum, Maximum, Sum and Count (number of
observations). You may ignore Kurtosis (a mathematical measure
of the concentration of data around the center compared to the tails
of the distribution) and Skewness (a mathematical measure of
the symmetry of the distribution). You may ignore Standard
Error and Confidence Level for now, but we will use
them in Module 1.4
The Descriptive Statistics tool provides everything we have discussed
then, except Q3 and Q1. To get these, we need to use the
Function feature of Excel. As for all Excel Functions,
position the cursor in a cell or select the cell in which you want
the quartile, for example M31. Select Insert from the
Standard Toolbar, then Function, then
Statistical (in left box of Paste Function dialog screen),
then Quartile (scroll down to find Quartile in right box of
dialog screen), then follow the dialog screen by inserting the cell
range for your data (e.g., L2:L30), and 1 for Q1. The resulting cell
formula is =QUARTILE(L2:L30, 1). Note that no labels are allowed so I
started the cell range with L2 rather than L1. You should get the
results in Worksheet 1.3.2. I repeated this process for the third
quartile, Q3, which I placed in M32. I also added the formula =(M32 
M31) in cell M33 to get the interquartile range. I added the labels
in cells L31, L32, and L33 for clarity since the function feature
does not insert the title of the function.
Worksheet 2
First Quartile, Q1 = 19 Third Quartile, Q3 = 22 Interquartile Range = 3
Lets use the above descriptive statistics to
completely describe the set of data.
Location Measures of Center and Spread
Fifty percent of the ordered observations fall below the median
of 21 days. Twentyfive percent of the ordered observations fall
below the first quartile of 19 days; and 75% of the ordered
observations fall below the third quartile of 22 days. The
interquartile range is 3 days from 19 to 22, representing the middle
50% of the ordered observations. The five number summary is complete
when we add the minimum value of 16 and maximum of 29 days.
Arithmetic Measures of Center and Spread
The mean is 21.07, or simply 21 days depending on the accuracy we
need for reporting purposes. The standard deviation is 2.94 or,
rounded, 3 days. The interpretation: most or 95% of the observations
fall within the interval: mean + 2 *s = 21 + (2 * 3) or
15 to 27 days. To use the Empirical Rule to completely summarize a
set of data which is approximately symmetric and bellshaped, and
having a mean of 21 and standard deviation of 3:
68% of the observations are within 21 + (1 * 3): 18 to 24 days.95% of the observations are within 21 + (2 * 3): 15 to 27 days.
100% of the observations are within 21 + (3 * 3): 12 to 30 days.
From the above, we would not expect any
observations below 12 days or above 30. Such observations would be
outliers.
Percentile
To find the percentile using Excel, select Insert on the
Standard Toolbar, then Function, then Statistical, then
Percentile, and then respond to the dialogue box by entering
the cell range for your numbers and the desired percentile. For
example, to get the 25th percentile for this illustration, the
resulting cell formula will be: =PERCENTILE(L2:L31,0.25). The
percentile will be placed in your spreadsheet wherever you have the
cursor (active cell).
The ZScore
What is the ZScore for the number 18?
ZScore = (number  mean)/standard deviation = (18  21)/3 = 1
So, 18 is one standard deviation to the left of the mean of 21. What is the ZScore for the number 24?
ZScore = (24  21)/3 = 1
The number 24 is one standard deviation to the
right of the mean. To get the ZScore from Excel, we use the
Standardize function. Position the cursor or point and click
in a cell in which you want to place the ZScore, such as Q2. Select
Insert from the Standard Toolbar, then Function, then
Statistical, then Standardize and respond to the dialog
box questions. You should replicate the cell formula
=STANDARDIZE(18,21,3) which gives a ZScore of 1. I actually prefer
to use the cell address for the number 18 so that I can copy the
standardize cell reference to standardize a whole row or column of
numbers. This would be =STANDARDIZE(L6,21,3). I then add a title to
the column, such as "ZScores." The fifth requirement in Project
Assignment 1 is to standardize (generate ZScores) for your column of
data.
ZScores are very handy ways of identifying outliers. Any ZScore
below 3 or above +3 would identify a number more than 3 standard
deviations from the mean. The sixth requirement in Project Assignment
1 is to remove any outliers from your data, and recompute the
descriptive statistics. Outliers are quickly identified by ZScores.
Removing an outlier from a real data set doesn't mean we can ignore
it for analysis  it just means we should analyze it separately from
the data.
A side note: to remove a data element from a column of numbers in
Excel means to remove it, not type over the number with a zero. To
remove a number, point and click to it, select Edit on the
Standard Toolbar, then Clear from the pulldown menu, then
select Contents. This creates a blank space where the number
once resided. If you want to remove the number and the space, select
Edit, then Delete from the pulldown menu, then check
Shift cells up if you have a column of data, or Shift cells
left if you have a row of data.
The Probability for the ZScore
Excel has a statistical function that automatically computes the
probabilities for ZScores. Let's find the probability of getting a
cycle time below 18 days (remember, time is a continuous variable, so
a number below 18 could be 17.9999). If you know the ZScore for a
number, the NORMSDIST function will give a cumulative
probability for the area under a symmetric bellshaped distribution
up to that ZScore. NORMSDIST stands for Standardized Normal
Distribution. Position the cursor or point and click in a cell in
which you want the probability. Select Insert, then
Function, then Statistical, then NORMSDIST and
simply enter the ZScore of 1. You should get 0.158655 or 0.16 or
16%. The probability of obtaining an observation less than 18 days is
0.16 or 16%. What is the probability of obtaining an observation
GREATER than 18? You guessed it: 100%  16% = 84%.
The seventh item in Project Assignment 1 is to compute the
probability of exceeding the 45th ordered (sorted) observation in
your data set. If you have more or less than 50 observations, compute
the process capability for the 5th from the last ordered
observation.
There are some alternative Excel procedures and manipulations, but we
have described the important basics. What if you want the probability
of getting a cycle time below 18 days but do not know the ZScore?
Use the Excel Function NORMDIST function. Point and click to a
cell where you want the probability, select Insert from the
Standard Toolbar, then Function, then Statistical, then
NORMDIST and respond to the dialog box questions by entering
the number, mean, standard deviation and the work TRUE). The cell
formula would look like this: =NORMDIST(18,21,3,TRUE).
The Normal Distribution
Before we close this set of notes with one more example, let me say a
few words about the normal distribution. The mean, standard
deviation, ZScores and normal probabilities or probabilities for
ZScores all depend on the distribution being approximately symmetric
bellshaped or "Normal". The distributions will never be perfectly
bellshaped in real life, but they don't have to be  approximately
normal is fine, especially if we have at least 30 observations. So,
how do you know if the distribution is normal? Here are some general
rules of thumb:
1). Check the histogram to see if it has a symmetric bell shape, without outliers (any observation with a ZScore greater than +3 or less than 3).
2). The mean should be approximately equal to the median. For this example, 21.07 is approximately equal to 21.
3). The interquartile range should be close to 1.33 times the standard deviation. Here, the IQR of 3 is fairly close to 1.33 times 2.94 (3.9) although we would like this somewhat closer.
4) Check theoretical ranges of the empirical rule and compare them to the actual data.68% of the data should be between 18 and 24. The actual count is 26 out of 30 or 80%.
95% of the data should be between 15 and 27. The actual count is 29 out of 30 or 96.7%.
100% of the data should be between 12 and 30. The actual count is 30 our of 30, or 100%.
Most of the rules of thumb work for this cycle
time example. Lets look at an example where there is an outlier.
Putting it Together with One
More Example
What if the 31st observation was 38? We already know that 100% of
the observations should fall within the range of 12 to 30, so 38 is
more than 3 standard deviations from the mean  do you remember what
we called that  yes, an outlier or a signal.
Process Control Chart
Suppose you created a process control chart to monitor the cycle
time process based on the original 30 observations. The mean of the
chart is 21, the upper control limit for a process is the mean plus
three standard deviations or 21 + (3 * 3) = 30. The lower control
limit is the mean minus three standard deviations or 21  (3 * 3) =
12. These are upper and lower control limits for processes since we
expect nearly 100% of the observations generated by a process to be
within the area of the mean + 3 Std Dev. The Upper
Specification Limit of 24 is set by the boss or the customer, or
whomever  it is not set by statistics. Here is the process control
chart:
Worksheet 3
Note that the 31st observation is above the UCL and would justify investigation as a signal that the process has gone out of control. Let's confirm this with finding the probability of getting an observation of 38 or greater, with a process mean of 21 and standard deviation of 3. First, the ZScore computation:
ZScore = (number  mean)/Std Dev = (38  21)/3 = 5.67
Now, find the probability using NORMSDIST function of Excel. The cell formula is:
=NORMDIST(5.67) = 0.99999999
Recall, this is the cumulative probability of
observing a number LESS than 38. So to get the probability of
observing a number greater than 38, take 1.0  .99999999 = 0.00000001
or there is a 0% probability of observing a 38 if the mean is 21 and
the standard deviation is 3  that's why its called a signal or
outlier.
Before leaving the process control chart, let's focus for a moment on
the upper specification limit set usually by the boss or customer.
The USL is 24. Way back at the beginning of this note set, Module
1.1, we indicated that there is no way we can please the boss with an
upper specification limit of 24. Now you should be able to say why
that is so. With a mean of 21 and a standard deviation of 3, we know
that 24 is +1 standard deviations from the mean ( z score = (24 
21)/3 = + 1). The probability of getting an observation above 24 is
found by first getting the probability of a number less than
24:
=NORMDIST(+1) = 0.84
This process is only capable of
satisfying the customer 84% of the time  that is a measure of
process capability. Now, to get the probability of finding a
number above 24: 1.0  0.84 = 0.16. There is a 16% probability of
getting an observation above 24  that is why the customer will not
be pleased. An upper specification limit one standard deviation above
the mean is a 1 sigma process  GE's goal is to have SIX SIGMA
processes  specification limits 6 standard deviations from the
mean.
Descriptive Statistics
What if you were just starting out and collected 31 observations
and had no prior knowledge of the mean, standard deviation, median,
IQR or histogram. You would first generate the histogram, as shown in
Worksheet 1.3.4 below.
Worksheet 1.3.4
By observation, we would say there is an
outlier to the right of the range of the data, thus making a
distribution that is skewed right. If we needed to report the
center and spread of the skewed distribution we would use the.......
that's right, the median and interquartile range. Here are the
descriptive statistics:
Worksheet 1.3.5
Time 

Mean 
21.61 
Standard Error 
0.75 
Median 
21 
Mode 
21 
Standard Deviation 
4.19 
Sample Variance 
17.58 
Kurtosis 
7.04 
Skewness 
2.22 
Range 
22 
Minimum 
16 
Maximum 
38 
Sum 
670 
Count 
31 
Confidence Level(95.0%) 
1.54 
1st Quartile = 
19 
3rd Quartile = 
22.5 
Interquartile Range = 
3.5 
Using this data, we see that the mean is now greater than the median, confirming the right skew. Note that the median remains 21, as with the original data set. The mean is larger since the outlier is pulling it to the right. Note also the standard deviation is 4.19 or 4.2 which is considerably inflated from the original data with its standard deviation of 3. Outliers always increase the standard deviation. To confirm that 38 is an outlier with this data, we standardize:
=STANDARDIZE(38,21.6,4.2) = 4.05
Thirtyeight is more than 3 standard deviations
from the mean. In a real project, we would segregate the 38 from the
rest of the data, investigate and make corrections, and redo the
descriptive analysis without the outlier. This is precisely what you
may have to do with your data in Project Assignment 1. Item 5 is to
standardize your data, Item 6 is to remove outliers and recompute
the descriptive statistics. Data without outliers is likely to be
symmetric, bell shaped if there was no bias in data collection.
Inferential statistics for the mean of a distribution, which includes
confidence intervals and tests of hypothesis, work best with
symmetric bell shaped distributions. These subjects are covered next
in Modules 1.4 and 1.5.
Optional Material on the
Mean
This module focused on the arithmetic mean, the typical measure of
central tendency, and the measure used in inferential statistics.
There are two other means I should briefly mention: the weighted
mean, and the geometric mean.
Weighted Mean
The weighted mean is a special case of
the arithmetic mean used when there are several observations of the
same value. Suppose you had cycle times of 21, 23, 19, 22, and 20,
which gives an arithmetic mean of 21 days. But suppose further that
cycle time 21 happened 20 times, 23 happened 5 times, 19 happened 50
times, 22 happened 10 times, and 20 happened 15 times. The mean cycle
time weighted by frequency of occurrence, or simply the weighted mean
is:
Weighted mean = sum of weights * observation / sum of weightswhere weights are frequency of occurrence in this example
Weighted mean = (21*20 + 23*5 + 19*50 + 22*10 + 20*15) / 100 = 20
Geometric Mean
The geometric mean applies when we are
interested in computing the mean for interest and growth rates,
percentages, and ratios (Mason, 1999). Suppose you have a CD paying
10% interest in Year 1 (I wish!), and then 4% interest in Year 2. The
arithmetic mean interest rate is
Mean interest rate, i = (0.10 + 0.04 ) / 2 = 0.07 or 7 %
However, to take into account the compounding effect, the geometric
mean gives a more precise outcome:
Geometric mean = (1 + i) = nth root [ (1+i) * (1+i) ]= square root [ (1+0.10) * (1+0.04) ] = 1.0696 or 6.958 %
Let's check this with $95,000:
95,000 * 0.10 = 9500 (Year One)
(95000 + 9500) * 0.04 = 4180 (Year Two)Total Interest = 9500 + 4180 = 13680
The geometric mean: 95000 * 0.6958 = 6610
(95000 + 6610) * 0.04 = 7070Total Interest = 6610 + 7070 = 13680
The arithmetic mean: 95000 * 0.07 = 6650
(95000 + 6650) * 0.07 = 7115Total Interest = 6650 + 7115 = 13765
The geometric mean gives the truer figure in recognition of the
compounding effect. In general: geometric mean = nth root
[x_{1}x_{2}...x_{n}] where all x
values must be positive.
A second application of the geometric mean is to find the percent
increase, or growth rate, over time. Suppose the University gave the
College of Business the following head count growth targets: 182 MBA
students in 2001 and 344 MBA students in 2005. The geometric mean
percent increase over time is:
= (n1) root of [value at end of period/value at beginning of period]  1
= (5 years  1) or 4th root of [344 / 182 ]  1
= .1725 or 17.25%
Reference:
Anderson, D., Sweeney, D., &
Williams, T. (2001). Contemporary Business Statistics with Microsoft
Excel. Cincinnati, OH: SouthWestern, Chapter 3 (except Section 3.5)
and Chapter 6 (Section 6.2).
Mason, R., Lind, D. & Marchal, W. (1999). Statistical
Techniques in Business and Economics (10th. ed.).
Boston: Irwin McGraw Hill, Chapter 2.


