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Stats #11: Computing Simple Descriptive Statistics

This class will teach you how to recognize different types of data, how to summarize your data, and how to display your data graphically. Please bring a copy of research paper comparing two groups (e.g., new versus standard therapy) for use in class exercises. This class is useful for anyone who needs to summarize or display numerical data. It is also recommended for anyone who needs to interpret simple descriptive statistics. There are no pre-requisites for this class.

In this class, you will learn how to:

  • recognize various types of data, identify limitations inherent in certain data types;
  • compute percentiles; and
  • construct and interpret a box plot;
  • characterize central tendency and spread of individual variables.

This class does not qualify for IRB Education Credits (IRBECs).

Contents

  • Overview of the STATS web pages
  • Consulting services that I provide
  • Definition: Qualitative data
  • Definition: Categorical data
  • Definition: Continuous data
  • Definition: Nominal data
  • Definition: Ordinal data
  • Definition: Interval data
  • Definition: Ratio Data
  • Definition: Stem and leaf diagram
  • Definition: Percentiles
  • Definition: Box plot
  • Definition: Sample mean
  • Definition: Median
  • Definition: Mode
  • Definition: Range
  • Definition: Standard deviation
  • Please fill out an evaluation form

Overview of the STATS web pages (January 21, 2000)

What are the STATS web pages?

The STATS pages are a collection of handouts that I use in my job as a statistical consultant. The web provides a nice home for these handouts, because as I update my material, the newest version is immediately available to anyone who is interested.

Where can I find STATS?

If you have a web browser, like Internet Explorer or Netscape Navigator, you can surf on over to my site,

http://www.childrensmercy.org/stats

which is also found at http://internet1/stats, if you are attached to the Children's Mercy Hospital network. There are two obsolete sites: http://www.cmh.edu/stats and http://simon/stats. Do not use either of these sites.

Some of the fun stuff you can find on the STATS web pages.

Ask Professor Mean.  For the tough Statistics questions that Dear Abby won't touch.

Planning Your Research Study.  Things you need to plan for before you start collecting your data.

Selecting An Appropriate Sample Size.  How much data do you really need?

Managing Your Research Data.  Everything you want to know before you step to the keyboard.

Steps In a Typical Data Analysis.  I have my data on the computer. Now what?

How to Read a Medical Journal Article.  Reading a journal is hard work. Here's some help.

Professor Mean's Library.  Good books and good web sites about Statistics.

... and even more good stuff!!!

This webpage was written, edited by Linda Foland, and was last modified on 07/08/2008. Category: Website details


For CMH employees only: Statistical Consulting Services.

You can get free statistical consulting if you work for Children's Mercy Hospital. Ashley Sherman provide a wide range of statistical consulting services to help you with your research projects. This help can start as early as the initial planning of your research. I also help with the analysis of your data, using SPSS or other statistical software. We can also provide assistance with the preparation of your presentations and publications.

Here area some examples of the services that we have provided:

  • setting up your research hypothesis,
  • selecting and justifying your sample size,
  • writing the statistical methods section for your grant,
  • preparing randomization tables for your study,
  • reviewing your surveys for content and quality,
  • developing a system for entering your data,
  • choosing an appropriate statistical model for your data,
  • establishing validity and/or reliability for your measurement scales,
  • checking for violations of statistical assumptions in your data,
  • producing graphs and tables for your research publication, and
  • providing references for new and unusual statistical methods.

Specific statistical advice has been outlined on a series of web pages which can be found at http://www.childrensmercy.org/stats/. The pages provide advice about planning your research, selecting an appropriate sample size, managing your research data, performing a variety of data analyses, presenting research data, and writing research papers.

This webpage was written on 2003-04-30 and was last modified on 2008-07-08. Category: Professional details


Directions to my new office (April 25, 2008).

I have moved to a new office. It is a modular building just north of Children's Mercy Hospital. It is between 23rd and 22nd street, just off of Kenwood Avenue (Kenwood is a small north/south street just west of Holmes). If you need to get from your office to mine, here are some directions written by my Administrative Assistant, Judy Champion.

  • Take the elevator of the research tower down to the yellow level. Exit the employee parking garage on 23rd Street, walk to Kenwood and cross 23rd Street. Your destination is Building M 3 which is the building closest to 22nd Street. However, the entrance to our building faces Building M 2. It's best to walk into the parking area that is just north of Building M 1 and follow the sidewalk around the west side of building M 2 in order to get to our building's entrance on its south side. Another route would be to exit the Hospital Hill Center Building on Holmes and then walk ' block north to 23rd Street, cross 23rd Street, walk west to Kenwood then north to building M 3 address 2220 Kenwood.

2008-07-14. Send Category: Professional details


What is qualitative data?

There are two separate definitions of this term. Some people use the term "qualitative data" to represent categorical data.

Other people use the term to represent verbal or narrative pieces of data. These types of data are collected through focus groups, interviews, opened ended questionnaire items, and other less structured situations.

Qualitative research is not the same as anecdotal information. Qualitative research is rigorous research with

explicit sampling strategies, systematic analysis of data, and a commitment to examining counter explanations. Ideally, methods should be transparent, allowing the reader to assess the validity and the extent to which results might be applicable to their own clinical practice. -- BMJ 1998; 316:1230-1232.

Denise Polit gives a delightful example of the difference between qualitative and quantitative data in her book Data Analysis and Statistics for Nursing Research.

Here is her example of qualitative data.

Have you felt sad or depressed at all lately, or have you generally been in good spirits?

(Subject 1) Well, I've been in pretty rough shape lately, to tell you the truth. I mean, I haven't felt suicidal or anything like that, but I just can't seem to shake the blues. I just don't see anything to feel hopeful about in my future. I haven't really had anybody to talk to about my problems since my husband died last year.

(Subject 2) I'm not at all depressed. I feel great! I love my new job. And I've lost 20 pounds and feel much healthier than I have in years. I can't remember any period of my life when I've been happier.

Here is her example of quantitative data.

Quantitative: Thinking about the past week, how depressed would you say you have been on a scale from 0 to 10, where 0 means "not at all" and 10 means "the most possible?"

(Subject 1) 9

(Subject 2) 0

This webpage was written on 2002-10-11 and was last modified on 2008-07-08. This page needs major revisions. Category: Definitions.


What is categorical data?

Data that consist of only small number of values, each corresponding to a specific category value or label. Ask yourself whether you can state out loud all the possible values of your data without taking a breath. If you can, you have a pretty good indication that your data are categorical. In a recently published study of breast feeding in pre-term infants, there are a variety of categorical variables:

  • Breast feeding status (exclusive, partial, and none);

  • whether the mother was employed (yes, no); and

  • the mother's marital status (single, married, divorced, widowed).

This webpage was written on 2002-10-11 and was last modified on 2008-07-08. This page needs major revisions. Category: Definitions.


What is continuous data?

Data that consist of a large number of values, with no particular category label attached to any particular data value. Ask yourself if your data can conceptually take on any value inside some interval. If it can, you have a good indication that your data are continuous. In a recently published study of breast feeding in pre-term infants, there are a variety of continuous variables:

  • the infant's birth weight in grams;
  • the mother's age in years; and
  • the distance from the mother's home to the hospital in miles.

This webpage was written on 2002-10-11 and was last modified on 2008-07-08. This page needs major revisions. Category: Definitions.


What is nominal data?

Nominal data are categorical data where the order of the categories is arbitrary. A good example is race/ethnicity has values 1=White, 2=Hispanic, 3=American Indian, 4=Black, 5=Other. Note that the order of the categories is arbitrary.

Certain statistical concepts are meaningless for nominal data. For example it would be silly to ask what are the mean and standard deviation are for race/ethnicity.

This webpage was written on 2002-10-11 and was last modified on 2008-07-08. This page needs major revisions. Category: Definitions.


What is ordinal data?

Ordinal data are categorical data where there is a logical ordering to the categories. A good example is the Likert scale that you see on many surveys: 1=Strongly disagree; 2=Disagree; 3=Neutral; 4=Agree; 5=Strongly agree.

While computation of a median is easily justified for ordinal data, some statisticians have reservations about computing a mean for ordinal data.

This webpage was written on 2002-10-11 and was last modified on 2008-07-08. This page needs major revisions. Category: Definitions.


What is interval data?

Interval data is continuous data where differences are interpretable, but where there is no "natural" zero. A good example is temperature in Fahrenheit degrees.

Ratios are meaningless for interval data. You cannot say, for example, that one day is twice as hot as another day.

Example: In a study of a seven item pictorial scale for three aspects of dyspnea (throat closing, chest tightness, and effort), children were asked to place these images on along a visual analog scale.

Results Children aged eight years or older rated the scales in the correct order 75% to 98% correctly, but children less than 8 years of age performed unreliably. The mean distance between each consecutive item in each pictorial scale was equal. Conclusion Preliminary results revealed that children aged 8 to 18 years understood and used these three scales measuring throat closing, chest tightness, and effort appropriately. The scales appear to accurately measure the construct of breathlessness, at least at an interval level. Additional research applying these scales to clinical situations is warranted. -- Dalhousie dyspnea scales: construct and content validity of pictorial scales for measuring dyspnea. Patrick J McGrath, Paul T Pianosi, Anita M Unruh and Chloe P Buckley. BMC Pediatrics 2005, 5:33doi:10.1186/1471-2431-5-33. [Medline] [Abstract] [Full text] [PDF]

This webpage was written on 2002-10-11 and was last modified on 2008-07-08. Category: Definitions.


What is ratio data?

Ratio data are continuous data where both differences and ratios are interpretable. Ratio data has a natural zero. A good example is birth weight in kg.

The distinctions between interval and ratio data are subtle, but fortunately, this distinction is often not important. Certain specialized statistics, such as a geometric mean and a coefficient of variation can only be applied to ratio data.

Example: In the discussion of the use of a Visual Analog Scale for present functioning of children or adolescents, researchers argued that the VAS possessed properties of a ratio scale.

Price et al. have investigated the measurement properties of the VAS in a series of psychophysical studies, demonstrating that the VAS has ratio rather than interval scale properties [29]. Thus, in addition to properties of interval scales which reflect equal distances in the variables being quantitatively ordered, the ratio scale indicates a true zero point [30]. -- The PedsQL' Present Functioning Visual Analogue Scales: preliminary reliability and validity. Sandra A Sherman, Sarajane Eisen, Tasha M Burwinkle, and James W Varni. Health and Quality of Life Outcomes 2006, 4:75doi:10.1186/1477-7525-4-75. [Medline] [Abstract] [Full text] [PDF]

This webpage was written on 2002-10-11 and was last modified on 2008-07-08. This page needs major revisions. Category: Definitions.


What is a stem and leaf diagram? (October 11, 2002)

A stem and leaf diagram provides a visual summary of your data. This diagram provides a partial sorting of the data and allows you to detect the distributional pattern of the data.

There are three steps for drawing a tem and leaf diagram.

  1. Split the data into two pieces, stem and leaf.
  2. Arrange the stems from low to high.
  3. Attach each leaf to the appropriate stem.

It's easiest to understand these steps through an example. Let's construct a stem and leaf diagram for the following data: PImax (cm H2O) for 25 patients with cystic fibrosis.

80, 85, 110, 95, 95, 100, 45, 95, 130, 75, 80, 70, 80, 100, 120, 110, 125, 75, 100, 40, 75, 110, 150, 75, 95

  1. Split the data into two pieces, stem and leaf. Here the leaf would be the single rightmost digit and the stem would be the leftmost one or two digits.
  2. Arrange the stems from low to high. Here the stems range from 4 to 15.
  3. Attach each leaf to the appropriate stem.
    80 -- Attach the 0 leaf to the 8 stem,
    85 -- Attach the 5 leaf to the 8 stem,
    110 -- Attach the 0 leaf to the 11 stem,...
    as so forth.

This is what you get when you are done.

04 50
05
06
07 50555
08 0500
09 5555
10 000
11 000
12 05
13 0
14
15 0

Notice that the stem and leaf diagram is also a sideways histogram.

The stem and leaf is also useful because it partially sorts the data. In this example, the third smallest PImax score is 75

Second example

Here is data set consisting of LDL (Low Density Lipoprotein) values (mmol/l) of 14 subjects on an oat bran diet:

3.84, 5.57, 5.85, 4.80, 3.68, 2.96, 4.41, 3.72, 3.49, 3.84, 5.26, 3.73, 1.84, 4.14

Let the first digit be the stem and the last two digits be the leaf. We could have made a different choic: letting the first two digits be the stem and the last digit be the leaf. Don't agonize over the choice, but it's good to have not too many and not too few stems.

When you arrange the leaves on the appropriate stems you get the following diagram.

1 84
2 96
3 84,68,72,49,84,73
4 80,41,14
5 57,85,26

Again, this is a partial sort of the data. The third smallest LDL value is 3.49.

Splitting stems

Sometimes you may have too few (or too many) stems to get a good picture of your data. When this happens, considering splitting the stems. Here is an example.

The stem and leaf diagram for the LDL data has only five stems. We can get a slightly different perspective by doubling the number of stems. We do this by splitting each stem in two. Put small leaves (00-49) on the first stem and large leaves (50-99) on the second stem. This is what you get.

1
1 84
2
2 96
3 49
3 84,68,72,84,73
4 41,14
4 80
5 26
5 57,85

Another option is to split the stems into five. Attach leaves 00-19 to the first stem, 20-39 to the second stem, 40-59 to the third stem, etc. Here is an example.

0
0
0 45,40
0 75,70,75,75,75
0 80,85,95,95,95,80,80,95
1 10,00,00,10,00,10
1 30,20,25
1 50
1
1

Splitting the stems into two or five pieces is optional. It just gives you extra choices for displaying your data.

A stem and leaf diagram in SPSS

To create a stem-and-leaf diagram in SPSS, select ANALYZE | DESCRIPTIVES | EXPLORE from the SPSS menu. Here is an example of the output.

Normal Oral Temperature Stem-and-Leaf Plot

Frequency Stem & Leaf

2.00 Extremes (=<96.4)
4.00 96 . 7789
13.00 97 . 0111222344444
21.00 97 . 556666777888888899999
38.00 98 . 00000000000111222222222233333444444444
33.00 98 . 555666666666677777777888888888899
15.00 99 . 000001112223344
2.00 99 . 59
1.00 100 . 0
1.00 Extremes (>=100.8)

Stem width: 1.0 Each leaf: 1 case(s)

Notice that SPSS tells you how many leaves are on each stem.

This webpage was written on 2004-03-05 and was last modified on 2008-07-08. This page needs minor revisions. Category: Definitions, Category: Descriptive statistics


What is a percentile?

The pth percentile is a value so that roughly p% of the data are smaller and (100-p)% of the data are larger. Percentiles can be computed for ordinal, interval, or ratio data.

There are three steps for computig a percentile.

  1. Sort the data from low to high;
  2. Count the number of values (n);
  3. Select the p*(n+1) observation.

You can't always be so lucky to have p*(n+1) be a nice whole number. Here are some contingencies.

  • If p*(n+1) is not a whole number, then go halfway between the two adjacent numbers.
  • If p*(n+1) < 1, select the smallest observation.
  • If p*(n+1) > n, select the largest observation.

Examples

The following data represents cotinine levels in saliva (nmol/l) after smoking. We want to compute the 50th percentile.

73, 58, 67, 93, 33, 18, 147

  1. Sorted data: 18, 33, 58, 67, 73, 93, 147
  2. There are n=7 observations.
  3. Select 0.50*(7+1)=4th observation.

Therefore, the 50th percentile equals 67. Notice that there are three observations larger than 67 and three observations smaller than 67.

Suppose we want to compute the 20th percentile. Notice that p*(n+1) = 0.20*(7+1)=1.6. This is not a whole number so we select halfway between 1st and 2nd observation or 25.5. (Some people see the 1.6 and think they have to go six tenths of the way to the second value. You can do this if you like, but I think life is too short to worry about such details.)

Suppose we want to compute the 10th percentile. Since 0.10*(7+1)=0.8, we should select the smallest observation which is 18.

The five number summary

A five number summary uses percentiles to describe a set of data. The five number summary consists of

  • MAX - the maximum value
  • 75% - the 75th percentile
  • 50% - the 50th percentile
  • 25% - the 25th percentile
  • MIN - the minimum value

The five number summary splits the data into four regions, each of which contains 25% of the data.

Example of a five number summary

Percentage of body fat was estimated for a random sample of 252 individuals. The five number summary is

MAX - 45.1
75% - 24.6
50% - 19.0
25% - 12.8
MIN - 0.0

The value of 0.0 is clearly in error. Either the formula for estimating percentage of body fat was applied incorrectly or the estimated percentage of body fat was intended to be coded as missing. With the 0.0 removed, the minimum value becomes 1.9.

This summary implies, for example, that a quarter of the sample had body fat percentages between roughly 25 and 45.

Another example of a five number summary

You might be curious about the types of people who were in the random sample described above. A five number summary shows a wide range of ages in this sample.

MAX 81
75% 54
50% 43
25% 35
MIN 22

Notice that these subjects are adults of all ages. The youngest quarter of the subjects range from 22 to 35 years in age. The oldest quarter range from 54 to 81 years.

Computing percentiles in SPSS

If you have a data set in SPSS, select ANALYZE | DESCRIPTIVE STATISTICS | EXPLORE to compute all the information you need for a five number summary.In the dialog box, be sure to click on the STATISTICS button and select the PERCENTILES option. An example of the output appears below.

Percentiles 5.0000 10.0000 25.0000 50.0000 75.0000 90.0000 95.0000

Haverage 25.0000 27.0000 35.2500 43.0000 54.0000 63.7000 67.3500

Tukey's Hinges 35.5000 43.0000 54.0000

This webpage was written on 2002-10-11 and was last modified on 2008-07-08. This page needs minor revisions. Category: Definitions, Category: Descriptive statistics.


What is a boxplot? (October 15, 2002)

The box plot is a graphical display of a five number summary. Sometimes the box plot is also known as a box and whiskers plot.

Here are the four steps you follow to draw a boxplot.

  1. Draw a box from the 25th to the 75th percentile.
  2. Split the box with a line at the median.
  3. Draw a thin lines (whisker) from the 75th percentile up to the maximum value.
  4. Draw another thin line from the 25th percentile down to the minimum value.

The length of the box in a box plot, i.e., the distance between the 25th and 75th percentiles, is known as the interquartile range. You can use this box length to detect outliers. If any whisker is more than 1.5 times as long as the length of the box, then we have evidence of outliers. A common variation on the box plot is to draw the whisker to the value which is just shy of 1.5 box lengths away, and highlight each individual data point more than 1.5 box lengths away.

This webpage was written on 2005-08-18 and was last modified on 2008-07-08. Category: Definitions, Category: Graphical display.


What is a mean? (sample mean/population mean)

The mean is simply the average of all the items in a sample. To compute a sample mean, add up all the sample values and divide by the size of the sample.

  • The cotinine values for seven smokers are: 73, 58, 67, 93, 33, 18, and 147. If you added up these values you would get a sum of 489. Divide that sum by 7 to get a mean of 69.9.

We will sometimes make the distinction between the sample mean and the population mean. The population mean (often represented by the Greek letter mu) is simply the average of all the items in a population. Because a population is usually very large, the population mean is usually an unknown constant. The formula for the population mean is:

wpeEA.gif (1083 bytes)

where N is the number of items in the population. Usually N is very large in the thousands, millions, or sometimes even infinity. The greek letter sigma indicates that you should add the values together.

The sample mean (often represented by the symbol XBAR) is the average of all the items in a sample. The sample mean is a lot easier to compute because the size of the sample is usually quite manageable. If the sample is chosen carefully, the sample mean is a good estimate of the population mean. The formula for the sample mean is

where n is the number of items in the sample.

This webpage was written on 2002-10-11 and was last modified on 2008-07-08. This page needs minor revisions. Category: Definitions, Category: Descriptive statistics.

 


What is a median?

The median is the value so that roughly half of the data are smaller and roughly half of the data are larger. There are two formulas for the computation of the median, depending on whether the size of your sample is even or odd. In both cases, sort the data. If n (the number of observations in your sample) is odd, select (n+1)/2 observation. If n is even, select halfway between the n/2 and n/2+1 observation.

  • In a sample of adults, the LDL values are (in order from low to high): 1.84, 2.96, 3.49, 3.68, 3.72, 3.73, 3.84, 3.84, 4.14, 4.41, 4.80, 4.26, 5.57, and 5.85. For this data there are an even number of observations (n=14). So we would select halfway between the 7th observation (3.84) and the 8th observation (also 3.84). Thus the median is 3.84 for this data set.

This webpage was written on 2002-10-11 and was last modified on 2008-07-08. This page needs minor revisions. Category: Definitions, Category: Descriptive statistics.

 


What is a mode?

The mode is the most frequently occurring value in the data set. Sometimes there is a bit of ambiguity in the selection of the mode. For example, two different values may both be tied for the most frequently occurring value. In this case, both values will be considered a mode of the data set.

Another situation where the mode is ambiguous is when no value in the data set occurs more than once. In this situation, any (and every) value is considered a mode.

A related statistics is the modal class. this is the most frequently occurring category when the data are classified into groups. The modal class, of course, will change depending on how wide or narrow your classifications are.

  • In a sample of adults, the LDL values are (in order from low to high): 1.84, 2.96, 3.49, 3.68, 3.72, 3.73, 3.84, 3.84, 4.14, 4.41, 4.80, 4.26, 5.57, and 5.85. For this data, the mode is 3.84, since that is the only data value which occurs twice in this data set. If we classified the data into groups like 1.00-1.99, 2.00-2.99, etc. the modal class would be 3.00-3.99.

This webpage was written on 2002-10-11 and was last modified on 2008-07-08. This page needs minor revisions. Category: Definitions, Category: Descriptive statistics.


What is a range?

The range (R) is the distance between the largest and the smallest numbers in the data.

The range is easy to understand and easy to compute. If you suspect, howver, that there are outliers in your data, then the range will be unduly influenced by those outliers.

Example

For the data below (LDL), the largest value is 5.85 and the smallest is 1.84.

1 84
2 96
3 84,68,72,49,84,73
4 80,41,14
5 57,85,26

The range is 5.85-1.84 = 4.01.

Compute the range for the Pimax data.

0
0
0 45,40
0 75,70,75,75,70
0 80,85,95,95,95,80,80,95
1 10,00,00,10,00,10
1 30,20,25
1 50
1
1

The largest value is 150 and the smallest is 40.

R = 150-40 = 110.

This webpage was written on 2002-10-11 and was last modified on 2008-07-08. This page needs minor revisions. Category: Definitions, Category: Descriptive statistics.


What is a standard deviation? (October 11, 2002)

The standard deviation is a measure of how spread out your data are. Computation of the standard deviation is a bit tedious. The steps are:

  1. Compute the mean for the data set.

  2. Compute the deviation by subtracting the mean from each value.

  3. Square each individual deviation.

  4. Add up the squared deviations.

  5. Divide by one less than the sample size.

  6. Take the square root.

Suppose your data follows the classic bell shaped curve pattern. One conceptual way to think about the standard deviation is that it is a measures of how spread out the bell is. Shown below is a bell shaped curve with a standard deviation of 1. Notice how tightly concentrated the distribution is.

Shown below is a different bell shaped curve, one with a standard deviation of 2. Notice that the curve is wider, which implies that the data are less concentrated and more spread out.

Finally, a bell shaped curve with a standard deviation of 3 appears below. This curve shows the most spread.

Example

Let's examine a standard deviation computation for data on PI max values in a sample of children with cystic fibrosis. The seven values in this data set are 73, 58, 67, 93, 33, 18, and 147. The mean for this data set is 69.9.

(73-69.9)2 = (3.1)2 = 9.61
(58-69.9)2 = (-11.9)2 = 141.61
(67-69.9)2 = (-2.9)2 = 8.41
(93-69.9)2 = (23.1)2 = 533.61
(33-69.9)2 = (-36.9)2 = 1361.61
(18-69.9)2 = (-51.9)2 = 2693.61
(147-69.9)2 = (77.1)2 = 5944.41

For each data value, compute the squared deviation by subtracting the mean and then squaring the result. The sum of these squared deviations is 10,692.87. Divide by 6 to get 1782.15. Take the square root of this value to get the standard deviation, 42.2.

Interpreting the standard deviation

For reasonably symmetric and bell shaped data sets:

Using SPSS

Select ANALYZE | DESCRIPTIVE STATISTICS | DESCRIPTIVES from the menu to get computation of the standard deviation.

This table shows that the standard deviation is 42.2.

Population standard deviation. The population standard deviation (often represented by the Greek letter sigma) is measures the variability of data in a population. It is usually an unknown constant. The formula is:

cint122a.gif (1260 bytes)

Sample standard deviation. The sample standard deviation (usually represented by S) measures the variability of data in a sample. It is easy to compute (compared to a population standard deviation) because it is based on a small and manageable sample. The formula is:

cint122b.gif (1280 bytes)

This webpage was written on 2002-10-11,n, and was last modified on 2008-07-08. Category: Definitions, Category: Descriptive statistics.


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