Question

Step 1

Find examples of measures of central tendency and variability.

Find an example of data presented in some print medium–newspaper, magazine, journal, etc.–and focus on the measures of central tendency and measures of variability.

Step 2

Write about your example.

Prepare your discussion posting by answering the following:

• Which measures or central tendency and variability were presented in the article?
• Were they presented correctly?
• Is there a clearer way the author could have presented the information?
• In what ways are the measures of central tendency and variability helpful to the reader?

Answer 1

Answer:

Central tendency

Definition: the tendency of quantitative data to cluster around some central value. The closeness with which the values surround the central value is commonly quantified using the standard deviation.

Measures of Central Tendency

1. Mean: The sum of all measurements divided by the number of observations.
2. Median: The middle value that separates the higher half from the lower half. Mean and median can be compared with each other to determine if the population is of normal distribution or not.
3. Mode: The most frequent value. Example of use: to determine the most common blood group.
4. Geometric mean - the nth root of the product of the data values.
5. Harmonic mean - the reciprocal of the arithmetic mean of the reciprocals of the data values.
6. Weighted mean - an arithmetic mean that makes use of weighting to certain data elements.
7. Truncated mean - the arithmetic mean of data values that do not include the whole set of values, such as ignoring values after a certain number or discarding a fixed proportion of the highest and lower values.
8. Midrange - the arithmetic mean of the maximum and minimum values of a data set.

   Variability (dispersion)

   Definition: dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions. It is the variability or spread in a variable or a probability distribution.[1]

   Measures of variability

9. Variance: A measure of how far a set of numbers are spread out from each other. It describes how far the numbers lie from the mean (expected value). It is the square of standard deviation.
10. Standard deviation (SD): it is only used for data that are “normally distributed”. SD indicates how much a set of values is spread around the average.[2] SD is determined by the variance (SD=the root of the variance).
11. Interquartile range (IQR): the interquartile range (IQR), is also known as the 'midspread' or 'middle fifty', is a measure of statistical dispersion, being equal to the difference between the third and first quartiles[3]. IQR = Q3 − Q1. Unlike (tota
12. l) range, the interquartile range is a more commonly used statistic, since it excludes the lower 25% and upper 25%, therefore reflecting more accurately valid values and excluding the outliers.
13. Range: it is the length of the smallest interval which contains all the data and is calculated by subtracting the smallest observation (sample minimum) from the greatest (sample maximum) and provides an indication of statistical dispersion[4]. It bears the same units as the data used for calculating it. Because of its dependance on just two observations, it tends to be a poor and weak measure of dispersion, with the only exception being when the sample size is large.

Measures of Central Tendency and Variability

Objectives

Upon completion of this assignment, you will be able to:

1. Describe the three major types of data.
2. List the three major measures of central tendency.
3. Given assorted distributions of numbers, be able to compute each of the three measures of central tendency.
4. Explain the differences between measures of central tendency as to their usage.
5. List three measures of variability.
6. Given assorted distributions of numbers, be able to compute each of the three measures of variability.
7. Explain the difference between measures of variability as to their usage.

Reading Assignments:

• Urdan, pp. 1-11, 13-18, and 19-28.
• Pyrczak and Bruce, pp. 91-96 and 97-106 (6th edition); 97-108 (7th edition); 97-107 (8th edition).
• Lang and Heiss, pp. 26-30.
• Isaac and Michael, pp. 164-165, and 172-173.
• Materials presented in this package.
• Tuckman, pp. 259-266.

Evaluation

1. Successful completion of this assignment can be accomplished by answering the following questions in typed format.
2. Define mean.
3. Calculate the mean for the following list of numbers:

   41	34	22	20	32	36	34	31	41	22	31	36	40	54	24
   22	8	33	42	41	33	23	30	41	39	27	41	28	34	26
   34	26	35	22	18	50	30	38	50	16	45	8	17	21	38
   36	38	35	39	40	24	21	36	35	43	36	26	50	47	36
   26	32	32	38	50	42	35	31	26	27	27	44	27	43	37

4. Define median.
5. Calculate the median for those numbers listed in question number 2.
6. Define mode.
7. Calculate the mode for those numbers listed in question number 2.
8. Cite the purpose for each measure of central tendency.
9. Which measure of central tendency would you employ in calculating an attitudinal survey?
10. Define variability.
11. Define range.
12. What is the range of the numbers listed in question 2?
13. Define variance.
14. Calculate the variance for the numbers listed in question 2.
15. Define standard deviation.
16. Calculate the standard deviation for the numbers listed in question 2.
17. Cite the purpose for each measure of variability.
18. Plot the standard deviation for this problem.

METHODS OF RESEARCH IN EDUCATION

OTED 635
Information Sheet

Types of Data

     Due to the differences in the way in which we measure things, the numbers we accumulate do not always mean the same thing. We still divide the numbers which researchers commonly encounter into three scales of data. Almost all numbers fall into one of these three data types.

INTERVAL

     Interval numbers are those numbers with which you are probably most familiar. As you can see from the number line, interval numbers are called so because the intervals between the numbers are equal.

                                                                               
-4   -3    -2    -1    0    1    2    3    4

     Some people call this type of data equidistant interval data. However, we simply call it interval data. Interval data is probably the most common type of data which we run into in educational research, the reason for this being that most standardized or classroom tests give us scores which can be interpreted as being interval in nature. For instance, if John gets a 20 on a test and Frank gets a 25 we assume the difference in these two scores to represent 5 units of knowledge or ability. Since these numbers do have equal intervals between them we can perform the usual arithmetic operations such as addition, subtraction, multiplication, and division. For this reason we can figure the mean with interval data. This is very important because being able to figure the mean is a first step in the use of most higher level statistical analyses.

ORDINAL

     Ordinal data are commonly encountered by researchers, but not as commonly as interval data. The most distinguishing characteristic of ordinal data is the fact that when placed on the line, the numbers do not have equal intervals between them. As you can see, although the numbers do get larger as we go to the right, we don't know how much larger they get.

                                                                               
    1           2   3       4

     Most ordinal data which is encountered by researchers comes to us in the form of ordinal ranks. Whenever we ask people to rank objects or attributes or other people in order of preference or in order on some scale, we are generating ordinal data. This is very important to remember.
     Ranks are ordinal data. Let us consider for a moment why ranks are ordinal. Suppose I asked you to rank 5 toothpastes in order of your preference for them. One of the toothpastes may be your all-time favorite, so you easily assign it "Rank 1." The other 4 toothpastes you may not be familiar with nor care for. However, you have heard of some more than others, so you place them in ranks 2, 3, 4, and 5. Now, it should be immediately obvious that the distance between ranks 2 and 3 for example, are not as great as the distance between ranks 1 and 2. Since the researcher does not know what goes on in the mind when objects are ranked, he does not know the intervals between people's ranks. Your jump from first to second place may be much larger than someone else's.
     Ordinal data then, come from ranks, but they cannot be treated as interval data because of the fact that they cannot be added, subtracted, multiplied, or divided. For this reason we cannot figure a mean with ordinal data. It is inappropriate to try to do so. If we need a measure of central tendency, we usually use the median. Since we cannot figure the mean with ordinal data, but still encounter numerous instances of ordinal data in educational research, statistics have been developed to work specifically with ordinal data doing tests similar to those that would be done with interval data. We call these statistics that have been developed for ordinal data "non-parametric statistics".

NOMINAL

     Nominal numbers themselves are seldom encountered in educational research. An example of a nominal number for instance, would be the numbers on a football jersey. The numbers mean nothing in terms of actual quantifications. They do, however, serve to label. For this reason we call them nominal numbers. Nominal data are usually encountered by researchers when we categorize things. For instance, we may count the number of apples, pears, peaches, and oranges produced by a certain farmer. Whenever we count things and then place them into categories, we are developing nominal data from a research point of view. One of the most common forms of nominal data which we encounter in research are the results of questionnaire studies. When we count the number of people who respond to Item A, Item B, Item C, etc.

MEASURES OF CENTRAL TENDENCY

     The behavior of an individual on a particular occasion may, or may not be typical of him. Therefore, it is often necessary to measure his behavior several times in order to get a good estimate of what he is likely to do. For example, suppose a man is being considered for an assignment as an astronaut. It is necessary to make certain he can react quickly to signals from the instruments in his space capsule. To test this, he is asked to press a button when a visual or an auditory signal is presented. His reaction time - the time required to press the button after the presentation of the stimulus - is then measured.
     Once a measurement of the astronaut's reaction time is obtained, the question is, "Does this reaction time represent his reaction time in general, or is he usually faster or slower?" He may have been slow in reacting because he was not paying attention. To determine what his typical reaction time is under the experimental conditions, it is necessary to measure several trials and determine the average time required to respond. It has been observed that when such a series of reaction times is obtained from a subject, most of the times tend to cluster about some central value, with some slightly longer and some slightly shorter than the more common ones. Statistics that describe the central value about which scores cluster are called measures of central tendency. A measure of central tendency can be thought of as the average score the individual would get in a large number of trials.

THE MEAN

     To illustrate the problem of obtaining a measure of central tendency, consider an experiment in which a subject moved a control knob whenever he heard a signal. The time between the presentation of the signal and the movement of the knob was measured for twenty trials. The times are given in the Table.

INDIVIDUAL REACTION TIMES

Trial		Reaction Time (milliseconds)		Trial		Reaction Time (milliseconds)
1		213		11		145
2		206		12		153
3		132		13		152
4		185		14		171
5		160		15		143
6		153		16		159
7		153		17		161
8		155		18		148
9		150		19		153
10		139		20		149

     Although there is quite a range of measurements for this subject, the times seem to cluster around 160 milliseconds. Some times are several milliseconds away from this value, but half are within 10 milliseconds of 160. This clustering of measurements would be more apparent if the raw data were arranged in a frequency distribution.
     One measure used to describe the central tendency of a group of scores is the arithmetic. The arithmetic mean is obtained by adding together all of the measurements and dividing by the number of measurements taken. In the example, the sum of the reaction times is 3180 milliseconds. When this total is divided by the number of trials (20), the mean is found to be 159 milliseconds.

COMPUTING THE MEAN

     To express in mathematical terms the method of finding the mean, let the letter X stand for the score on each trial. Thus X1 could stand for the time of the first trial, X2 the time on the second trial, X5 the time on the fifth trial, and so on to Xn the nth or last score obtained. The arithmetic mean is designated by (read " bar X") and the formula for finding the mean is:

This formula says that, regardless of how many observations are made, the mean is always equal to the sum of the values for all of the observations, divided by the number of observations. The formula presented above is inconveniently long to write, so it is usually given in a briefer form in which the instruction to add together all of the X's is indicated by the symbol for summation, . The formula then becomes:

     One must always remember that X means "the sum of all the X's". Substituting values from the reaction-time example, this formula yields:

or 159 milliseconds

What has been said about finding the mean reaction time of one subject can also be said about finding the mean reaction time of a group of individuals. In the latter case, the mean for each individual is represented by X value. Thus, the mean time for the first subject (S1) is represented by X1, the mean for S2 by X2, and the mean for the nth subject Xn.

Suppose the mean reaction times of twelve subjects have been obtained by the method used in the earlier example. The means are shown in the Table:

Mean Reaction Times for Twelve Subjects

Subject

Mean Reaction Time (milliseconds)

Subject

Mean Reaction Time (milliseconds)

A	160	G	152
B	162	H	159
C	178	I	163
D	154	J	160
E	148	K	165
F	158	L	161

     Here, the we want is the mean score of the twelve individuals who, in turn, had mean scores as shown in the table. The same formula is used for finding the mean of the group of subjects as for finding the mean scores of each of the individuals:

     N = 12

     That is, the mean score of a group of individuals is obtained by dividing the sum of their scores by the number of individuals in the group.

THE MEDIAN

     A second measure of central tendency is the median. The median is the value above which half of the measure lie. It is used when the distribution is badly skewed -- when the measures are piled up at one end of the distribution rather than being more or less symmetrically distributed about the mean. The difference between symmetrical and skewed distributions can be seen below. When a distribution is skewed, the median is a better indicator of the central point about which most of the scored cluster.

Symmetrical	Skewed

Symmetrical and Skewed distributions

     The achievement test scores from a class of thirty students, some of whom had been promoted without being prepared, appear below:

Achievement Test Scores

80	70	67	62	59	36
77	69	67	62	57	33
75	69	66	61	53	20
74	68	65	61	51	19
70	67	64	60	47	13

     The mean of this distribution is 58, but two-thirds of the students did better than this. The median is 63, since half of the scores are above this value. In this example the median is more indicative of typical performance of the unprepared students.
     There is a formula for finding the exact value of the median in cases when it is not as obvious as in the example. However, it is often sufficiently accurate just to rank the measures from high to low (as was done in the example) and count down to the middle of the distribution.
     When the middle, or mid-point score, falls between two numbers, as in this case, 64 and 65, we add the two numbers together and divide them by two and we get our median as 63. Another example would be to have our median fall between 58 and 61. What would the mid-point be? It would be 59.5.

THE MODE

     A third measure of central tendency is the mode. The mode is the score that most frequently occurs. In the distribution of achievement test scores given above, the mode is 67 because more subjects had this score than any other. The mode is seldom used in statistics, but it is a quickly found measure of central tendency, since it can be easily identified without computation. In those cases where it typically is required, the mode provides the necessary information.
     The concept of modality is useful in describing a distribution that has two very frequent scores, with less frequent scores appearing between them. When this occurs in large distributions it usually means that scores of two unlike groups have been combined in a single distribution. For example, animals of the same species but of different strains often behave differently on the same task. These differences may be masked if their scores on the task are combined -- except that the combined distribution may have two modes, with the scores on one group clustering around another mode. Such distributions are called bimodal; that is, they have two modes.

     The following table lists the number of errors made by twenty rats that learned a maze.

Errors in Learning a Maze

4	8	11	13
6	8	12	14
7	9	13	15
8	9	13	16
8	10	13	18

     The two modes of this distribution are at 8 and 13.

SUMMARY

     We have then, three different measures of central tendency. Each measure is used for different purposes, and has different limitations as to use.

MEAN

     The mean is best used to describe the average of a distribution of numbers that is fairly symmetrical. That is, a distribution without extreme scores at one end. Because it is used so much in statistics, the mean is the most important measure of central tendency for statisticians. One limitation is that the mean must be figured from interval or ratio data. We cannot take the mean of ordinal numbers, since we don't know how far the numbers are from each other in terms of distance. Most experiments involve comparing mean behavior, or mean performance between different groups.

MEDIAN

     The median is best used to describe the middle of a distribution of numbers which has extreme values at one end or the other. Remember that the median is not affected by extreme scores as is the mean. A very common use of the median is to evaluate the central tendency of ordinal data (such as ranks), since we cannot compute the mean of such data.

MODE

     The mode is seldom used in statistics. If it is used, the mode is usually found for one of these reasons:

1. To determine whether or not a distribution is bimodal.
2. In specific cases, it is interesting to find the most often appearing score in a distribution of scores.

MEASURES OF VARIABILITY

     The mean, or some other measure of central tendency, is useful in determining what is representative behavior for an individual or group. It does not provide answers to such questions as, "How well does the measure of central tendency represent the group?" To answer these questions it is necessary to consider how the values in a distribution vary from one another and from the mean. This "spread" or dispersion in a set of measurements is called variability.

Range

     The simplest measure of variability is the range. The range is the difference between the two extreme values in the distribution. Consider a distribution of scores (representing trials to an errorless performance) made by a group of girls on a finger maze: 3, 5, 6, 8, 9, 11, 12, 18, and 14. The range is equal to the highest value minus the lowest value: 14 - 3 = a range of 11.
     The range is easily calculated but tells us little about the variability of values within the distribution. Compare the distribution of scores made by a group of boys on the same finger maze - 3, 7, 8, 8, 9, 10, 11, 11, 14 - with the distribution above. These distributions have the same range and mean, but it is obvious that most of the measurements cluster more closely about the mean in the second distribution than in the first. The amount by which values deviate from the mean is greater in the first distribution. This example makes it clear that the range does not describe the variability in a group of measurements well enough to clearly distinguish one group from others in which the measures are distributed differently. Thus we need a measure of variability that will take into account every measurement rather than just the highest and lowest.

VARIANCE

     A measure of variability will describe a group of measurements better if it is based on every measurement in the group. It is also useful to give special weight or importance to those measurements that are the most unusual -- that is, those measurements farthest from the mean of the distribution. These conditions for a method of describing variation have been met in a statistic called the variance. The variance is defined as the mean of the squares of the deviations of each measurement from the mean. The formula for the variance (s2) is :

where

2	=	the sum of the squared values of all the scores
X	=	the sum of all the scores
n	=	the number of scores

The variance includes all the data, because every measure in the group is used in its computation. Special weight is given to the extreme values because the deviation of each score is squared. The variance is a kind of average which reflects the distance of the individual scores from the mean of the distribution. The larger the variance, the greater the variability; that is, the greater the distance of scores from the mean. The smaller the variance, the less the variability.

COMPUTING VARIANCE

EXAMPLE

     This example is drawn from the area of development psychology. Assume that an experimenter is interested in obtaining information about the heights of twelve-year-olds. To do this, he collects a sample of data (the heights of twenty randomly selected twelve-year-olds) and proceeds to compute the variance.

Step l.    Table the data as follows. No particular order is necessary.

Subject

Height (inches)

Subject

Height (inches)

Subject

Height (inches)

Subject

Height (inches)

S1	64		S6	59		S11	60		S16	55
S2	48		S7	57		S12	43		S17	56
S3	55		S8	61		S13	67		S18	64
S4	68		S9	63		S14	70		S19	61
S5	72		S10	60		S15	65		S20	60

Step 2.    Add all the scores. (Note: If you are using a calculator you can do Step 2 and Step 3 at the same time.)

64 + 48 + ... + 60=1208

Step 3.    Square all the scores and add the squared values.

642 + 482 + ... + 602= 73,894

Step 4.    Square the sum obtained in Step 2, and divide this value by the number of scores that were added to obtain the sum. The resultant value is called the correction term.

Step 5.    Subtract the value obtained in Step 4 from the sum in Step 3.

73,894 - 72,963 = 931

Step 6.    Divide the value obtained in Step 5 by N - l.1 (In this example, it would be 20 - 1 = 19). The resultant value is the variance.

STANDARD DEVIATION

The most commonly used measure of dispersion of a group of scores is not the range, nor the variance, but another statistic called the standard deviation. The standard deviation of a group of scores is an index of the degree to which the scores do or don't cluster around the mean. If we have a loose distribution with most scores right around the mean. The standard deviation is used quite often in statistics because it has certain mathematical properties which are quite useful in describing a group of scores.

COMPUTATION

     Computation of the standard deviation from a distribution of numbers is quite simple. In fact, if you know how to find the variance, you are almost done finding the standard deviation. The reason for this is that the standard deviation equals the square root of the variance.

     Therefore, to find the standard deviation of a group of scores, find the variance and take its square root. To find the standard deviation of our previous example, we would take the square root of the variance, which was 49, giving us a standard deviation of 7

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