C-1: Variables which are measured on a continuous scale are described in terms of three key characteristics central tendency, variability, and shape.

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1 MODULE 02: DESCRIBING DT SECTION C: KEY POINTS C-1: Variables which are measured on a continuous scale are described in terms of three key characteristics central tendency, variability, and shape. C-2: One of the most common quantities used to describe a set of data is its center. In the example above, your GP (Grade Point verage) was used to describe the "center" or "middle" of your scores or grades in individual courses. In statistics, the term central tendency refers to the middle or center of the data on any given variable. There are many descriptive statistics which are use to identify the "center" of a group of scores, and the appropriate one to use depends on the variable and the way it was measured. C-3: The most familiar of the descriptive statistics used to convey information about a variable's central tendency is the arithmetic average or mean. The mean is computed by adding together all of the individual scores on the variable, and dividing the total by the number of scores. Using your grade point average as an example, let's assume that you have taken 5 courses and have earned 3 's, 1 B, and 1 C in these courses. To compute your grade point average, we would add 4 points for each earned, 3 points for each B earned, and 2 points for each C earned. Thus, your total points earned would be computed as TOTL POINTS = =17. To compute your GP, we take the total points earned and divide by the number of courses. In this example, you have taken 5 courses, so your GP is 17/5= 3.4. Since your grade point average is the B range (assuming that a B is anywhere from 3.0 to 3.5) you would be carrying a "B" average, be described as a "B" student, and be expected to earn a "B" in most of the courses you take. However, in looking at your individual scores, you have actually earned more 's than any other grade! Does it seem right that you would be described as a "B" student? Part of the problem comes from the fact that your course grade is really not a continuous measure of your performance. warding students an, B, C, D, or F in a course reduces their performance to a more ordinal measure. In doing this, we lose the difference between a "High " and "Low." Because of this problem, the mean is usually only used as a measure of central tendency for variables which are measured on a continuous scale. nother problem comes from the fact that your course grades are kind of lopsided - almost skewed in their shape. The mean is not a good descriptive statistic to use when the distribution of the variable is not symmetrical. In order to describe variables which are discrete, or which are nonnormal in shape, we need to use a different descriptive statistic. The final problem with the mean is that it is really affected by extreme scores. Considering our example of your GP, one F in one class will really pull your GP down! The mean is very sensitive to extreme scores, whether they are 's or F's. It is only a good measure of central tendency if the data set does not include extreme values (high or low). Page 1 of 7

2 C-4: the median is the numeric value which divides your list of scores in half, with lower scores in the bottom half and higher scores in the top half. To find the median, we arrange of the scores in order, from lowest to highest, and then find the value in the middle. This value is the median. If we have an odd number of scores (i.e. our sample consists of an odd number of patients) then finding the middle number is easy. For example, if we take a body weight measurement of a sample of 7 different patients, and arrange their weights in order from lowest to highest, our resulting data set would look like this: 122 lbs 137 lbs Median = 137 lbs Mean = (122 lbs lbs )/7 = 10836/7 = lbs The median of this data set is 137 lbs, because that is the middle weight, with 3 lower weights below it, and 3 higher weights above it. In this case there is no computation required at all! However, if we have an even number of patients (e.g. n = 6) we might have to do a bit of computation to get the median. Take the same list of body weights as shown above, but remove the lowest weight (122 lbs). The median of the resulting data set is computed as the sum of the middle two values (137 lbs + = 285 lbs) divided by 2 or lbs as shown below: 137 lbs Median = (137lbs + )/2 = lbs The median is a better descriptive statistic to use than the mean when the variable is measured on an ordinal scale because it is based on the order of the scores, not on their actual values. Let's go back to your hypothetical GP. If we arrange your scores from low to high we get: C B In this case, the grade is the median, because it divides the grades into two groups, with two lower grades below it, and two higher grades above it. Since you actually scored more 's than any other score, this seems a bit more accurate when describing your grades, doesn't it? The best option here would be to report both the mean (3.4) and the median (). Using both descriptive statistics, we have characterized your academic performance a bit more accurately than we could have done with either statistic alone. Page 2 of 7

3 It is can also be a helpful companion to the mean when we are describing a continuous variable which is skewed or when there are extreme values in the data set. In either case, the median is not "pulled" or influenced as the mean, and so it may be a better descriptor of the center of the distribution. Let's take our patient weights, only this time let's add a very heavy patient to the original 7 scores. 122 lbs 137 lbs Median = (137 lbs + 148lbs)/2 = 285/2 = lbs 420 lbs Mean = (122 lbs lbs lbs)/8 = 1503/8 = lbs With the new heavier patient, the median moves a bit from 137lbs to 142.5lbs. But the mean is really affected by the extreme weight, moving from 154.7lbs to187.9lbs! For this reason, when scores are skewed or where there are outliers present, the median should be used along with the mean to describe the central tendency. C-5: The mode is the simplest descriptive statistic which is used to measure central tendency. The mode is defined as the most frequently occurring score in a data set. Therefore, to compute the mode we simply need to arrange the scores on our variable in order and select the value which occurs most often. The mode is typically only used to describe scores on variables which are measured on a categorical scale. Let's begin with your GP this time. If we arrange the scores in order, we see that the most frequently occurring grade in the data set is an. Thus the mode is defined as. C B There are 3 's, but only 1 B and only 1 C, so is the mode because it occurs most frequently. Page 3 of 7

4 With variables measured on the ordinal or nominal scale, the mode works very well. However, with continuous variables it may not work at all. Taking our 7 patients and their body weights as an example, we quickly see that this data set has no mode - or since all values occur exactly once, perhaps it actually has 7 modes! 122 lbs 137 lbs In the case of continuous variables like body weight, the mode is often not a very helpful descriptive statistic when we are interested in central tendency. In this case, we are better off using the mean and/or median to describe the middle of the data set. C-6: While the measures of central tendency provide information about what individuals have in common (e.g. the average of their test scores), the measures of variability quantify the degree to which they differ. If not all values of data are the same, they differ and variability exists. The descriptive statistics used for variability describe how "spread out" individual scores are. The larger the statistic, the more "spread out" the individual scores are. The terms variability, spread and dispersion all mean the same thing. C-7: If we know the central tendency of our set of scores that is a good start. However, it is not enough to fully describe our data. descriptive statistic for variability is necessary because different groups, or different samples, vary in different ways. For example, suppose that a teacher gave the same test to two different classes and obtained the following scores. Each class has 5 students as follows: Class 1: 80%, 80%, 80%, 80%, 80% Class 2: 60%, 70%, 80%, 90%, 100% If you calculate the mean for both sets of scores, you will get the same answer: 80%! However, it is obvious that the two sets of scores are not the same. The scores in Class 2 obviously vary more than the scores in the other class. We must use a measure of variability to describe this difference, and to distinguish between these two classes. In this case, the mean is just not enough. Page 4 of 7

5 It is important to describe both the central tendency and the variability of a variable if we want to be able to communicate or compare variables or samples. It may be important to know how two different groups compare on their "center" and on their "spread." For example, if we want to compare the body weight of male and female patients, it would be helpful to know if the average age for males was higher or lower than that for females. It might also be helpful to know if the weights of the male patients varied more than the weights of the female patients. C-8: The easiest and most familiar measure of variability is called the range. The range is simply the lowest value in our data set subtracted from the highest value in our data set. If we used the body weights of our 7 patients as an example again, the range would be computed as RNGE=-122 lbs = 98 lbs. 122 lbs lowest score 137 lbs highest score Range = highest score - lowest score = lbs = 98 lbs The range usually works best in describing continuous variables. But like the mean, the range is really affected by extreme values. If we add our 420 lb patient to the data set, the range increases from 98 lbs to 420 lbs- 122 lbs = 298 lbs! That much of a change just because one score has changed makes our statistic very unstable. It also means that all of the score in between the lowest and the highest don't even matter when we think about how spread out our scores are. That makes the range a very imprecise measure, because all of the scores are not included when we calculate it. Page 5 of 7

6 C-9: The variance of a set of scores is defined as the "average squared distance" of the scores from the mean. To compute the variance, we must first measure how far each data point is from the mean. That is pretty easy - we just take each observed score and subtract the mean. For our 7 patient weights, that would look something like this: 122 lbs lbs = lbs lbs = lbs lbs = lbs 137 lbs lbs = lbs lbs = -6.7 lbs lbs = 40.3 lbs lbs = 65.3 lbs fter I find the difference between each observed score and the mean, I square each difference like this: 122 lbs lbs = lbs (-32.7 lbs)(-32.7 lbs) = lbs lbs = lbs (-29.7 lbs)(-29.7 lbs) = lbs lbs = lbs (-18.7 lbs)(-18.7 lbs) = lbs lbs lbs = lbs (-17.7 lbs)(-17.7 lbs) = lbs lbs = -6.7 lbs (-6.7 lbs)(-6.7 lbs) = 44.9 lbs lbs = 40.3 lbs (40.3 lbs)(40.3 lbs) = lbs lbs = 65.3 lbs (65.3 lbs)(65.3 lbs) = lbs 2 Finally, I add up all of the squared differences, and divide by the total number of patients -1. In this case, the denominator is 7-1 = 6. var = (1069.3lbs lbs lbs lbs lbs lbs lbs 2 )/(7-1) = 8,547.5 lbs 2 / 6 = lbs 2 The variance, usually symbolized as s-squared, is equal to lbs 2. This means that the average squared distance of the weights in the data set from the mean is lbs 2. The larger the variance, the greater the variability in the scores. This estimate of the variability of the scores is more precise than the range, because it includes each of the scores in the data set in the Page 6 of 7

7 computation. But what is a squared pound? We will need a measure of variability which is more understandable, and not in squared units. C-10: the standard deviation, defined as the average distance of the scores in the data set from the mean is the most frequently used measure of variability for continuous variables. It is computed pretty simply by taking the square root of the variance. Based on our weight example, that means: standard deviation (s) = lbs 2 = 37.7 lbs The standard deviation, usually symbolized as s, has the advantage of being reported in the original units! No square pounds are necessary, so it is much easier to understand. s we saw with the variance, the larger the actual value, the greater the variability in the scores. C-11: So far we haven't really talked about a measure of variability that works well with categorical variables. Using our GP example, we would want to express the variability in the following grades: C B We can accurately say that the scores vary from "C" to "." We could even assign point values like we did before, giving all of the 's a 4, B's a 3, and C's a 2. But we still only really have data measured on an ordinal scale. Even using the range by subtracting the lowest from the highest score (4-2 = 2) gives us the number 2. In our example 2 is equivalent to a "C" - so does that mean we have a range of "C?" Here, whether or not we assign numeric variables to the categories of our ordinal variable, they are only categories. There is not much to be gained by trying to describe the variability in the scores with a descriptive statistic. Our verbal description is really the best. Page 7 of 7

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