Descriptive statistics, part II: most commonly used descriptive statistics. (Scientific...

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In order to promote evidence-based

practice, pediatric nurses need to read and critique relevant research. Most research reports provide some descriptive information about the sample and data collected. A wide variety of descriptive statistics is used in research, but the nurse needs to know which statistics should be used for different kinds of data. It is especially important to know which statistics are appropriate for data of differing levels of measurement. The most common descriptive statistics fall into one of the four groups listed in Table 1 (Larson & Farber, 2002).

Measures of Shape

The distribution of a variable is the way that its data cluster or spread across its continuum. It is the shape of the data when the data am graphed so that the levels of the variable are on the x-axis and the number of cases found at each data point constitutes the y-axis (Figure 1). The shape for ordinal or interval/ratio variables that is most useful for most data analysis testing is a normal distribution, often called the bell-shaped curve (Figure 2). A normal distribution for ordinal and interval/ratio data allows the use of a wider variety and more powerful statistics. Skewness and kurtosis are two descriptive statistics that test the normality of the shape of the distribution.

Tests of Skewness

Skew measures whether the two halves of the distribution are symmetrical (Glass & Hopkins, 1996). A distribution that has a long tail on the right is called positively skewed (Kiess, 2002), and might be what is seen with a chart for survival of a childhood cancer, where very few children die from a disease early but, over time, an increasing percentage die (Figure 3). The opposite, with a long tail on the left side of the graph, is a negative skew. Either positive or negative skew indicates a non-normal distribution. Tests of skew in SPSS require both skew and standard error of the skew. The researcher must divide the skew by its standard error. A normal distribution is represented by a value that ranges from [+ or -]2. Values much lower than -2 or higher than +2 denote a skewed rather than normal distribution.

Tests of Kurtosis

A distribution should be nicely and evenly rounded--neither too peaked, like a pencil sticking up from the graph, nor too flat, like a low hill. A distribution that is narrow from side to side and very peaked (high in the middle) is called leptokurtotic ("lepto' means thin). The opposite of leptokurtotic is platykurtotic, which refers to a flattened distribution. In a platykurtotic distribution, the sides are wide and the top is low in the graph. Both platykurtotic and leptokurtotic distributions are non-normal. A normally curved distribution is called mesokurtotic (Schmidt, 1975). Similar to skew, the kurtosis statistic in SPSS must be calculated from both the kurtosis and standard error of kurtosis. And the result must be between [+ or -]2.

Measures of Central Tendency

Measures of central tendency, sometimes called measures of location, are useful in identifying where the bulk of the cases fall in the distribution. They help the researcher determine where the most common or typical cases are most likely to fall (Sprinthall, 2003). Different levels of measurement (see Part I [McHugh, 2003]) require different measures of central tendency. And it is important to understand that the shape of the distribution is also influential in the choice of a descriptive statistic.

The Mode

The first measure of central tendency is the mode. It is the only useful measure of central tendency when a variable is measured on a nominal scale. A typical example of use of the mode to describe nominal data is illustrated in the following example:

   In an office practice, the pediatric-nurse practitioner
   (PNP) might well keep track of the final diagnosis
   of 3,000 children who came in during the
   winter months with the following symptoms: clear
   or slightly runny nose, fever, generalized aching of
   the muscles of the body, and either lethargy or irritability.
   The PNP might have found that the final
   distribution was as seen in Figure 1. It can be seen
   that by far the most common diagnosis was that
   the child had a cold. This is the central tendency in
   diagnosis for that constellation of symptoms.

How could such a finding be used? One of the most common uses of such data is in training clinicians about how to make diagnosis decisions. An important diagnostic concept is what one should look for first when a particular constellation of signs and symptoms is presented. As the old saying goes, "If you hear the pounding of hooves across the Kansas plains, you should think first of a herd of horses, and last of a herd of zebras." What this means is, know the most common illnesses in your population, and look for them first. Only after the clinician rules out a common illness should the clinician search for rare or exotic illnesses.

An important use of the mode when looking at a variable in a research study is to see how evenly distributed the cases are among all the levels. When a nominal variable represents grouping cases into various experimental and control groups for a clinical trial, it is important that the groups be roughly equal in size. Some of the most powerful statistics that can be used to test differences among study groups (i.e., ANOVA, ANCOVA, and MANOVA) will tolerate some size differences among the groups. But when very large group-size differences exist, different and less powerful statistics that adjust for unequal group sizes must be used.

The Median

The median is the exact midpoint, or 50th percentile, of a distribution. The median of a distribution, such as the distribution of various severities of acne in high school children shown in Figure 2, is calculated taking the highest scoring child and putting him or her on the far right, and the lowest scoring child and putting him or her on the far left. That process is continued until you find the exact middle child in the distribution. For ordinal data, the median is usually the best indicator of central tendency. In a normal distribution, the mean, median, and mode will all have the same value. However, in a skewed distribution, they may have very different values. In that case, the median is usually the most accurate indicator of central tendency. Figure 2 shows an example of an ordinal distribution. It represents the results of nurses' evaluations of acne severity in a population of teenage children in the high school. In the fictional sample represented in Figure 2, the mean is 3.2, which is closest to the numeric code of the "moderate" category, which is 3. The exact midpoint (median) is case number 49, whose score was moderate. The mode also can be seen to be moderate since the largest number of cases (22) scored at that level.

Compare that to Figure 3, which shows a very different distribution. Just by looking at the distribution, one can see that the mode and median of this distribution are probably different. In fact, the mode is the 5th year, while the median is located in the 7th year, and the mean is between the 7th and 9th years. The skew toward the left distorts the central tendency such that mean, median, and mode have somewhat different values, which would not happen in a normal distribution.

The Mean

The mean is a very powerful measure of central tendency. Its strength is, paradoxically, also its weakness. It incorporates the exact score from every subject into its estimate of central tendency. Thus, it can be used in all sorts of mathematical manipulations and statistical analyses. When some subjects' scores are extreme, however, the mean is distorted. That is, it is made artificially high or low by extreme scores that drag the mean away from the middle such that the mean is not a good representation of the central tendency of the variable. The median is not affected by the value of even the most extreme scores. It does not incorporate their actual scores, only the fact that there is a case out there. Thus, extreme scores do not distort the median. But the median cannot be used in the most powerful statistical analytic techniques.

The Problem of Extreme Scores

Extreme scores have a powerful effect on the mean when the sample size is small. They can be a problem even in a fairly large sample if several scores am very extreme. For example, in union negotiations for employee wages, it is typical for management to want to use the mean salary, and unions to insist on using the median salary to represent typical worker wages. This is because today many corporations annually pay their chief executive officer a salary and stock options in the millions of dollars. Regular hourly employees, however, seldom earn more than $50,000 annually. Because the top management salaries are so high, the mean is strongly distorted by the very few extremely high salaries. Thus, the mean salary is much higher than a typical factory worker could ever earn. For example, if 1,000 employees earn $40,000 each, two top executives each earn $10 million, and the chief executive earns $20 million, the median salary is $40,000, but the mean salary is $79,670--almost $80,000 per year, and double what the average worker earns.

Politically, this plays out as follows: Management knows that public opinion would not support a raise for factory workers in a company where the "mean" salary is "$80,000." Thus, they publicize information that emphasizes that the mean salary is already $80,000 a year in their factory; and ask why a company paying such a high mean salary should award any raises--especially in a tight economy. The union, however, knowing that the mean salary includes the millions paid to the people at the very top, does not like to use the mean. Instead, it uses the median or mode salary in advertisements so that the figure will be much closer to the typical factory worker's salary. As a result, the public doesn't know what to believe and believes both sides am lying. Yet, neither side is lying. In a skewed distribution, wide differences between the mean, median, and mode am quite likely to be found.

In summary measures of central tendency must be selected and used with great cam so that the test used provides the most accurate image of the most representative cases in the distribution. Nominal measures always use the mode. Ordinal measures most typically use the median but, if normally distributed, may use the mean. Interval and ratio measures most typically require the mean. If there is serious skewness, however, a variable measured at the interval or ratio levels may well be better represented by the median or even the mode.

Measures of Dispersion or Variation

Another issue researchers consider is variability of the scores among the subjects. Variability describes how cases tend to be scattered throughout the entire range of the variable. For a nominal variable, the only way to measure that concept is to look at the way cases are spread out among the possible categories of the variable. This is done by examining a graph of the data to see how many categories have subjects, and how many subjects there am per category. But the mathematical properties of ordinal, interval, and ratio scales allow for more precise measurement of variability. The measures of variation or scatter include range, variance, and standard deviation.

Range

The least precise measure of scatter is the range. The range of a variable means the distance from the lowest to the highest actual values scored (Larson & Farber, 2003). It is calculated by subtracting the score of the lowest scoring case from the score of the highest scoring case (Glass & Hopkins, 1996). This lets the researcher know how widely the subjects actually scored across the possible range of values. It does not, however, tell the researcher if cases were evenly spread across the possible values or if there were clumps of cases and empty spots on the scale. Although no measure perfectly describes scatter, the variance and standard deviation provide excellent descriptions of average variability in the dataset.

Variance and Standard Deviation

If we subtract each score from the mean of the variable, we have an indication of how much each of the scores deviates from the mean (Table 2). Unfortunately if we try to sum those scores, they always sum to zero, and that isn't mathematically helpful. So it is necessary to square the deviations to convert them all to positive numbers (e.g., -[2.sup.2] = +4) so that they will not sum to zero (e.g., -2 + 2 = 0). The sum of the squared deviations can be divided by the sample size to obtain an average deviation, and that is the variance. It is very specific to the particular distribution, however, and it is still burdened with all those squares. The measure is now in squared units rather than the original units, which is not very useful. For example, the variance of a variable that consists of the height in inches of a number of 16-year-old boys is expressed in square inches. Thus, it is not the original measure and its utility is limited.

A better option is to take the square root of the variance to eliminate the effect of squaring all those numbers (Glass & Hopkins, 1996). When the square root of the variance is obtained, the result is the standard deviation (represented by the Greek letter delta: [delta]). This is an extremely useful statistic; it is expressed in the original unit of measurement, and it provides a reliable estimate of the degree to which the numbers in the variable deviate from the mean. A small standard deviation (relative to the mean) means that most of the scores cluster tightly around the mean.

It turns out that one [delta] above and below the mean of any distribution represents approximately 68% of the cases (Figure 4). Ninety-five percent of the subjects will fall within 2 6 above and below the mean of the distribution. And 99% fall within [+ or -]3 [delta]. Thus, the standard deviation is truly a standard measure of variability that applies to any distribution, regardless of the unit of measure used (Larson & Farber, 2003). If the standard deviation is very small, scores are not scattered far from the mean. The larger the standard deviation, the more widely scattered are values in the distribution.

[FIGURE 4 OMITTED]

Percentile and Quartile Measures

Sometimes it is useful to know where a particular subject's score falls relative to the entire distribution. An excellent statistic for this purpose is the percentile. A percentile orders all the scores from highest to lowest and calculates the percentage of scores that fall below each of the individual scores (Loether & McTavish, 1974). Most national school achievement tests are reported to parents in terms of the percentile in which the child's score falls. For example, if a child received a mathematics score of 66, this means that 66% of the children scored lower on the test than he or she did. By simple subtraction, then, it is easy to know the percentage of cases that scored above the selected score. In the previous example, 33% of the children scored higher than the child who scored in the 66th percentile.

A quartile is simply the entire percentile chart divided into four equal sections. Scores from the 75th to the 99th percentile form the top quartile. The 50th to the 74th percentile form the second highest quartile. Scores from the 26th to the 49th percentile form the third highest quartile, and scores from the 1st to 25th percentile form the lowest quartile. One way to describe central tendency for such a distribution is to describe the scores in the second and third quartile, which is called the interquartile range (Glass & Hopkins, 1996). This interquartile range is the box of a box and whisker plot (Figure 5).

[FIGURE 5 OMITTED]

Summary

Descriptive measures can reveal a great deal of information about any variable of interest, whether the data be clinical, administrative, educational, or research data. To make best use of a descriptive statistic, it is important to know what levels of measurement should be used with the statistic, and what information the statistic can provide. To find out about the most typical case, measures of central tendency are appropriate. To discover whether the variable has a normal distribution, measures of shape should be applied. And to discover the variability about the mean of the variable, measures of dispersion should be used. Finally, percentiles and quartiles are useful for describing the placement of a single case in a population.

Table 1. Most Commonly Used Descriptive Statistics

        Different Types of Descriptive Statistic Measures

Shape Form, or     Central Tendency  Measures of         Quartile and
Normality          (also called      Dispersion          Percentile
Statistics         Location          or Variation        Measures
                   Statistics

Skew (symmetry of  Mode              Range               Percentile
the distribution)

Kurtosis           Median            Variance            Interquartile
("peakiness" or                                          range
flatness of the
distribution)

                   Mean              Standard Deviation


Table 2. Obtaining the Variance and Standard
Deviation of a Set of Numbers

          Subject's        Score-   (Score-
            Score    Mean   Mean   Mean) (2)

Mike       10        8.25   1.75     3.06
Jean       12        8.25   3.75    14.06
George      3        8.25  -5.25    27.56
Susan      15        8.25   6.75    45.56
Alice       8        8.25   -.025    0.06
Mary        4        8.25  -4.25    18.06
Tom         9        8.25   0.75     0.56
Joan        5        8.25  -3.25    10.56
  Totals   66 (a)           0      119.50 (b)
Range     3-15 = 12

(a) The scores sum to 66. The mean is achieved by dividing the total
scores by the number of students (66/8 = 8.25).

(b) The sum of squares (SOS) is 119.50. The variance is achieved by
dividing the SOS by the mean (119.50/8.25 = 14.48). The standard
deviation is the square root of the variance
([square root of 14.48] = 3.8).

Figure 1. Distribution of a Nominal Variable (N = 3,000)

Percent distribution of infections illnesses seen
in the Ill Child Clinic

Levels of the Variable

Encephalitis    3
Influenza      20
Common cold    64
Chicken pox     9
Mumps           4

Note: Table made from bar graph.

Figure 2. Approximately Normal Distribution,
Ordinal Variable

Incidence of acne in teenaged children

None          5
Mild         10
Mild-Mod     18
Mod          22
Mod-Severe   16
Severe       12
Polycystic    4

Note: Table made from bar graph.

Figure 3. Positive Skew (to the Right), Interval
Variable

15-year survival rates, Pediatric Cancer Center

 5th yr   70
 7th yr   65
 9th yr   50
11th yr   35
13th yr   25
15th yr    5

Note: Table made from bar graph.

References

Glass, G., & Hopkins, K. (1996). Statistical methods in education and psychology (3rd ed.). Boston: Allyn and Bacon.

Kiess, H. (2002). Statistical concepts for the behavioral sciences (3rd ed.). Boston: Allyn and Bacon.

Larson, R., & Farber, B. (2002). Elementary statistics: Picturing the world (2nd ed.). Upper Saddle River, NJ: Prentice Hall.

Loether, H., & McTavish, D. (1974). Descriptive statistics for sociologists. Boston: Allyn and Bacorr

McHugh, M. (2003). Descriptive statitics, Part 1: Level of measurement. JSPN, 8, 35-37.

Schmidt, M. (1975). Understanding and using statistics: Basic concepts. Washington, DC: Heath.

Sprinthall, R. (2003). Basic statistical analysis (7th ed.). Boston: Allyn and Bacon.

Search terms: Descriptive statistics

Mary L. McHugh, PhD, RN

Associate Professor, School of Nursing

University of Colorado Health Science Center Denver, CO

Author contact: mary.mchugh@uchsc.edu, with a copy to the Editor, roxie.foster@uhsc.edu