Internet Exercises 1. Find the two articles attached by M. L. McHugh. In the first article(Part 1 Article), examine Figure 2 and its associated narrative. Briefly explain why the variable in that figu

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 differ- ent kinds of data. It is especially important to know which statistics are appropriate for data of differing lev- els 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 are 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 dis- JSPN Vol. 8, No. 3, July-September, 2003111 tribution, often called the bell-shaped curve(Figure 2). A normal distribution for ordinal and interval/ratio data al- lows the use of a wider variety and more powerful statis- tics. 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 distri- bution are symmetrical (Glass & Hopkins, 1996). A dis- Scientific Inquiry Descriptive Statistics, Part II: Most Commonly Used Descriptive Statistics Mary L. McHugh Column Editor: Diane Hudson-Barr Table 1. Most Commonly Used Descriptive Statistics Different Types of Descriptive Statistic Measures Shape, Form, or Central Tendency Measures of Dispersion Quartile and Percentile Normality Statistics(also called Location Statistics)or Variation Measures Skew (symmetry of Mode Range Percentile the distribution) Kurtosis (“peakiness” Median Variance Interquartile range or “flatness” of the distribution) Mean Standard Deviation Figure 1. Distribution of a Nominal Variable (N= 3,000) 70 60 50 40 30 20 10 0 32064 9 4 Encephalitis Influenza Common Chicken Mumps cold pox Levels of the Variable Percent distribution of infectious ill- nesses seen in the Ill Child Clinic 112JSPN Vol. 8, No. 3, July-September, 2003 tribution that has a long tail on the right is called posi- tively 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 ±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 nonnor- mal. 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 stan- dard error of kurtosis. And the result must be between ±2.

Measures of Central Tendency Measures of central tendency, sometimes called mea- sures of location, are useful in identifying where the bulk of the cases fall in the distribution. They help the re- searcher determine where the most common or typical cases are most likely to fall (Sprinthall, 2003). Different levels of measurement (see Part I [McHugh, 2003]) re- quire different measures of central tendency. And it is im- portant 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 vari- able 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 practi- tioner (PNP) might well keep track of the final di- Scientific Inquiry Figure 2. Approximately Normal Distribution, Ordinal Variable Incidence of acne in teenaged children 25 20 15 10 5 0 5101822 16 12 4 None Mild Mild– Mod Mod– Severe Polycystic Mod Severe Figure 3. Positive Skew (to the Right), Interval Variable 15- year survival rates, Pediatric Cancer Center 80 70 60 50 40 30 20 10 0 70 35 65 25 5 50 5th yr 7th yr 9th yr 11th yr 13th yr 15th yr JSPN Vol. 8, No. 3, July-September, 2003113 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 ordi- nal data, the median is usually the best indicator of cen- tral tendency. In a normal distribution, the mean, me- dian, 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 accu- rate indicator of central tendency. Figure 2 shows an ex- ample 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 sam- ple 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 differ- ent 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 be- tween the 7th and 9th years. The skew toward the left distortsthe 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 ten- dency. Its strength is, paradoxically, also its weakness. It incorporates the exact score from every subject into its es- timate 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 dragthe 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 af- fected by the value of even the most extreme scores. It does not incorporate their actual scores, only the fact that agnosis 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 irri- tability. 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 tendencyin 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 diagnos- tic concept is what one should look for first when a par- ticular 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 clini- cian rules out a common illness should the clinician search for rare or exotic illnesses.

An important use of the mode when looking at a vari- able in a research study is to see how evenly distributed the cases are among all the levels. When a nominal vari- able 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 un- equal 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 114JSPN Vol. 8, No. 3, July-September, 2003 In summary, measures of central tendency must be se- lected and used with great care so that the test used pro- vides the most accurate image of the most representative cases in the distribution. Nominal measures always use the mode. Ordinal measures most typically use the me- dian but, if normally distributed, may use the mean. In- terval 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 bet- ter 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 are per category. But the mathematical properties of ordinal, interval, and ratio scales allow for more precise measure- ment 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 scor- ing case (Glass & Hopkins, 1996). This lets the re- searcher know how widely the subjects actually scored across the possible range of values. It does not, how- ever, 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 mea- sure perfectly describes scatter, the variance and stan- dard deviation provide excellent descriptions of aver- age variability in the dataset. Scientific Inquiry there is a case out there. Thus, extreme scores do not dis- tort 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 are very ex- treme. 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 execu- tive 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 man- agement salaries are so high, the mean is strongly dis- torted 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 mil- lion, and the chief executive earns $20 million, the me- dian 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 empha- sizes 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 me- dian 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 are lying. Yet, neither side is lying. In a skewed distribution, wide differences between the mean, median, and mode are quite likely to be found. Variance and Standard Deviation If we subtract each score from the mean of the vari- able, 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 2= +4) so that they will not sum to zero (e.g., – 2 + 2 = 0). The sum of the squared devia- tions 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 num- ber 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 vari- ance 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: δ). This is an ex- tremely 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 δabove 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 δabove and below the mean of the distribu- tion. And 99% fall within ±3 δ. Thus, the standard devia- tion is truly a standard measure of variability that ap- plies 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. JSPN Vol. 8, No. 3, July-September, 2003115 Percentile and Quartile Measures Sometimes it is useful to know where a particular subject’s score falls relative to the entire distribution. An Figure 4. Standard Deviations and Percent of Distribution Standard Deviation (δ) 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 Tom9 8.25 0.75 0.56 Joan 5 8.25 – 3.25 10.56 Totals 66 a 0119.50 b Range 3–15 = 12 aThe scores sum to 66. The mean is achieved by dividing the total scores by the number of students (66/8 = 8.25).

bThe 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 ( √14.48 = 3.8).

0 5 10 15 20 25 30 Mean 1 _ = 68% 2_’s = 95% 3_’s=99% 3 δ = 99% 2 δ = 95% 1 δ = 68% 116JSPN Vol. 8, No. 3, July-September, 2003 Scientific Inquiry excellent statistic for this purpose is the percentile. A percentile orders all the scores from highest to lowest and calculates the percentageof scores that fall below each of the individual scores (Loether & McTavish, 1974). Most national school achievement tests are re- ported 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 chil- dren scored lower on the test than he or she did. By sim- ple subtraction, then, it is easy to know the percentage of cases that scored above the selected score. In the previ- ous example, 33% of the children scored higher than the child who scored in the 66th percentile.

A quartile is simply the entire percentile chart di- vided 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 high- est 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 in- terquartile range (Glass & Hopkins, 1996). This in- terquartile range is the box of a box and whisker plot (Figure 5). Summary Descriptive measures can reveal a great deal of infor- mation 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, mea- sures of central tendency are appropriate. To discover whether the variable has a normal distribution, measures of shape should be applied. And to discover the variabil- ity about the mean of the variable, measures of disper- sion should be used. Finally, percentiles and quartiles are useful for describing the placement of a single case in a population.

Mary L. McHugh, PhD, RN Associate Professor, School of Nursing University of Colorado Health Science Center Denver, CO Author contact: [email protected], with a copy to the Editor, [email protected] References Glass, G., & Hopkins, K. (1996). Statistical methods in education and psy- chology(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 Bacon.

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 Box and Whis ker Chart 99 75 0 50 25 Highest Score Lowest ScoreMedi an Score Middl e 50% of Cases 75% 25% Figure 5. Box-and-Whisker Chart Showing Various Percentile Levels Highest score 75% Middle 50% of cases Median score 25% Lowest score