A staff member of a city parks and recreation department wanted to collect data to determine how people might differ in their use of a city park in differen

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LESSON 2: ANALYZING QUANTITATIVE DATA: DESCRIPTIVE STATISTICS (from Drew et al., 2008)

Once the data have been collected and raw scores tabulated, a researcher usually has several

operations to perform. In nearly all cases, the first process is data compilation in some form that

describes the group (the collection of participants’) performance. Compilation or data summary in

descriptive form involves the use of descriptive statistics. Descriptive statistics are tools for data

analysis that allow the researcher to determine how well the participants performed on a task, or

scored on a measure. Descriptive statistics permit the researcher to see how much variation there was

in the group – that is, whether many of the participants scored above the average or just a few. This

type of summary is very helpful because it tells the researcher far more about his or her participants’

performance than just examining a listing of individual scores or seeing that a particular individual

answered all but two of the questions correctly. Descriptive statistics describer how the participants in

the study behaved or performed.

Two general categories of descriptive statistics are commonly used – central tendency measures

and dispersion measures. Central tendency measures provide an index of where the scores tend to

bunch together or the typical score in the group of scores. Dispersion measures, on the other hand,

describe the amount of variability among the scores in the group. Measures of central tendency and

dispersion are the ones frequently used in graphs and tables.

If the primary intent of the study is to address a descriptive question, the researcher may well end

analysis with the computation of descriptive statistics. Such statistics will provide the information

necessary for the goal of group description. It is possible, however, that the investigator’s purpose

goes beyond describing the group. Perhaps the researcher wants to determine if some participants

perform better than others on a task (difference question), or if one aspect of their performance is

related to another (relationship question). Such a study would then go beyond group description. If

this is the case, descriptive statistics would allow preliminary computations necessary to perform

further data analysis using inferential statistics (discussed below). Descriptive statistics are commonly

computed regardless of whether the study addresses a descriptive, relationship, or difference research

question. In the first case, the descriptive statistics are probably the end points of the analysis,

whereas in the latter cases they are preliminary steps in preparation for further analysis. Entire

sequences of courses are designed to teach inferential statistics so we won’t even approach the topic

in the short time we have in this course. We will, however, show you how you can at least describe

your data to provide an accurate picture of it, whether you have a descriptive, relationship, or

difference research question utilizing descriptive statistics.

A. Measures of Central Tendency

Data may be summarized and presented in a number of ways, Frequently, it is desirable to be able

to characterize group scores with a single index that will provide some idea of how well the group

performed, or how they felt. This requires a number that is representative of the group of scores

obtained. One type of index commonly sued for such group description involves numbers that reflect

the concentration of scores, known as central tendency measures. Three measures of central tendency

are generally discussed: the mean, the median, and the mode. Each measure has slightly different

properties and is useful for different circumstances.

The Mean

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Probably the most familiar measure of central tendency is the mean. To obtain the mean, or

arithmetic average, simply add together all of tmeanhe participants’ scores and divide this total by the

number of individuals in the group. The mean of a group of scores is denoted with a capital letter with

a line across the top. For example, the mean of a group of scores on variable X is shown as X, and

read as “X bar”; the mean of the scores on variable Y, or “Y bar”, would be shown as Y.

The Median

A second measure of central tendency is the median. The median is a point in the distribution that

has exactly the same number of scores above it as below it when all scores are arranged in order. The

specific point at which the median exists in a given distribution is slightly different depending on

whether the number of individuals in the group (N) is odd or even. If N is odd, then the midpoint is

the middle score after the scores have been put in ascending or descending order. For example, if the

scores are 2, 17, 3, 29, 8, the median would be equal to 8 – which is the middle score after rearranging

their order from highest to lowest (or lowest to highest). If N is even, the median is a hypothetical

score midway between the two scores that occupy the middle position in the distribution. For

example, in the set of scores 86, 12, 19, 7, 44, 62, the median is between the two middle scores (19

and 44). To find the median value, take the average of the two middle scores (add them together, then

divide by 2). For this example, the median would be 31.5 [ = (19 + 44) / 2]. The symbol denoting the

median is Md or Mdn.

The Mode

The third measure of central tendency is the mode. The mode is simply an indicator of the most

frequent score – that is, more participants obtained that score than any other. For example, in the set

of scores 3, 17, 108, 14, 2, 5, 17, 12, the mode is 17 because it is the score that occurs more than any

other. If there is one mode, the distribution is called “unimodal.” For any group of scores, there may

be more than one mode. There may be two modes (called a bimodal distribution), more than two

modes (called a multimodal distribution), or no mode (when all scores appear with equal frequency).

B. Measures of Variability / Dispersion

Describing a set of scores with central tendency measures furnishes only one description of a

distribution, where the scores tend to be concentrated. In performing this function, the central

tendency measure, whether it is the mean, median, or mode, attempts to characterize the most typical

score with a single number. A second important way of describing scores involves measures of

dispersion. Dispersion measures provide an index of how much variation there is in the scores – that

is, to what degree individual scores depart from the central tendency. By determining where the

scores concentrate and to what degree individual performances vary, a more complete description of

the distribution is provided. In fact, in the absence of a dispersion measure to accompany the central

tendency, knowledge about any set of scores is limited. Four measures of dispersion are typically

found in research and each provides somewhat different information about dispersion: the range,

semi-interquartal range, variance, and standard deviation.

The Range

The range is the simplest and most easily determined measure of dispersion. As suggested by the

name, the range refers to the difference between the highest and lowest scores in a distribution. To

determine the value of the range in a set of scores, you simply subtract the highest score minus the

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lowest score. For example, if the scores were 12, 58, 6, 19, 44, 72, 31, the range would be 66 (72 – 6).

This example illustrates how easily the range in a set of scores may be determined. The range

provides a quick index of variability, which gives additional information beyond the central tendency

measure. Unfortunately, the usefulness of the range is somewhat limited because it uses little of the

data available in a set of scores. It is determined entirely by the two extreme scores. Since the extreme

scores may be highly variable, the range may fluctuate a great deal. Its usefulness is mostly limited to

preliminary data inspection. Extreme scores may be due to a number of factors that are not very

representative of the participant’s usual performance (e.g., physical or emotional upset, fatigue, or

irregularities in circumstances).

The Semi-Interquartile Range

The second measure of dispersion is the semi-interquartile range. The use of this measure

circumvents some of the difficulties noted with using the range. Quartiles are points on a

measurement scale that serve to divide a distribution of scores into four equal parts. The semi-

interquartile range is half the range in scores represented by the middle 50% of the scores. Look at the

figure below to examine this definition more closely.

Q Q2 Q3

The middle 50% of the scores in this distribution represents those scores between Q1 and Q3. To

establish the first and third quartiles, a simple counting procedure is involved. Q1 is determined by

counting up from the bottom of the distribution until a fourth of the scores have been encountered. If

there were 48 participants in the total group, than a fourth of that would be 12. The 12th score up from

the lowest would be at the first quartile (Q1). Similarly, the 12th score down from the highest would be

at the third quartile (Q3). Midway between these two points would be the second quartile, Q2. Note

that since Q2 is the middle-most score in the distribution, it is also the median. The semi-interquartile

range is represented by Q3 minus Q1 divided by two, or

Q3 – Q1

= Semi-interquartile range

2

For example, if Q1 was at 36 in the hypothetical distribution of scores and Q3 was at 92, then the

semi-interquartile range would be 92 – 36 = 56/2 = 28. The semi-interquartile range is generally

represented by Q, therefore in the example Q = 28.

The Variance

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The third measure of dispersion is the variance. By far the most commonly used index of

variability, the variance involves somewhat more complicated computational procedures. The

variance may be thought of as a measure of variability in the scores around the mean. The mean is

therefore the reference point, and the variance provides a description of the distribution of individual

scores around that point. In fact, the actual deviation of each score from the mean (mean – X) is used

in calculating the variance. The variance, in one sense, might be thought of as an average of all of the

deviations from the mean. The variance is expressed in score units and represents a width index along

the measurement scale. The width of this index becomes greater when the score are more variable and

narrower as the scores are more concentrated around the mean. Hence, if you were comparing one

group of scores where the variance was 12.6 to another group of scores where the variance was 8.9,

you would conclude that the higher variance indicates that there was less consistency (more

variability) among the scores in the first group than in the second.

The Standard Deviation

An additional measure of dispersion is also quite frequently used, and is called the standard

deviation. The standard deviation is simply defined as the square root of the variance, hence, you will

need to calculate the variance in order to determine the standard deviation. It is simply a measure of

convenience – by taking the square root of the variance you are simply reducing the size of the

dispersion measures while maintaining their meaning and relationship to one another.