Central Tendency & Variability

1. Measures of Central Tendency
   1. Mean
   2. Median
   3. Mode
2. Measures of Variability
   1. Range
   2. Mean Deviation
   3. Variance
   4. Standard Deviation
3. Estimation
4. Overall Example - [Minitab]

Homework

I. Measures of Central Tendency (or Averages)

Here, we are interested in the typical, most representative score. There are three measures of central tendency that you should be familiar with. Note that when reporting these values, one additional decimal of accuracy is given compared to what is available in the raw data (even if the additional decimal is a zero, e.g., 43.0).

1. Mean
   It is simply the arithmetic average or sum of the scores divided by the number of them.
   It is symbolized as:
   (read as "X-Bar") when computed on a sample.
     (read as "Mew") when computed on a population.

   Computation - Example:
   
   

   X
   2
   3
   5
   10
   åX = 20
   N=4

   
   Since means are typically reported with one more digit of accuracy that is present in the data, I reported the mean as 5.0 rather than just 5.

   When working with grouped frequency distributions, we can use an approximation:

   

   For example:

   Interval	Midpoint	f	Mid*f
   95-99	97	1	97
   90-94	92	3	276
   85-89	87	5	435
   80-84	82	6	492
   75-79	77	4	308
   70-74	72	3	216
   65-69	67	1	67
   60-64	62	2	124
   		åf=25=N	åMid*f=2015

   

   When computed on the raw data, we get:

   

   Thus the formula for computing the mean with grouped data gives us a good approximation of the actual mean. In fact, when we report the mean with one decimal more accuracy than what is in the data, the two techniques give the same result.

   Properties
   
   

   1. It is sensitive to all of the scores. In other words, if one score in the distribution is changed, the mean will change too. Example:
      
      

      Xs
      1, 2, 3	2
      1, 2, 30	11
      1, 2, 300	101

      
      
   2. The sum of the deviations about the mean equals zero. A deviation is symbolized as c (little x) and refers to the difference between a score and its mean. That is:
      
      
      Thus, this second property of the mean states that:
      
      
      What follows is some sample data demonstrating this property.
      
      

      X
      2	5	-3
      3	5	-2
      5	5	0
      10	5	5
      åc = 0

   3. The sum of the squared deviations about the mean is less than the sum of the squared deviations about any other value. Example (with "4" as the arbitrary "other value"):
      
      

      X	c	c2	X-4	(X-4)2
      2	-3	9	-2	4
      3	-2	4	-1	1
      5	0	0	1	1
      10	5	25	6	36
      åc2= 38			å(X-4)2 = 42

      So, 38 is less than 42. This relationship would hold with any "other value."

2. Median or M_d
   The score that cuts the distribution into two equal halves (or the middle score in the distribution).

   Computation - There are several situations possible:

   1. An odd number of scores and no duplication near the middle, then the median is the middle score.  
      Ex: 1, 2, 2, 4, 6, 7, 7.   N=7 & M_d= 4.

   2. An even number of scores and no duplication near the middle, then the median is the average of the two middle scores.  
      Ex: 2, 2, 4, 6, 7, 7.   N=6 & M_d = (6+4)/2 = 5.

   3. Duplication near the middle.  
      Ex: 4, 5, 5, 5, 6, 6.   N=6 & M_d = ?
      

      
      
      

      • So the median is somewhere between 4.5 and 5.5.
      • The lower exact limit of the score near the middle that is duplicated is 4.5 and we need 2 of the 3 scores in the interval with the duplication.
      • Thus, M_d = 4.5 + 2/3 = 4.5 +.67 = 5.17

   Fortunately, there is a formula to take care of the more complicated situations, including computing the median for grouped frequency distributions.
   
   

   
   
   Where:

   L	= Lower exact limit of the interval containing Md.
   nb	= number of scores below L.
   nw	= number of scores within the interval containing Md.
   i	= the width of the interval (for ungrouped data i=1).
   N	= the Number of scores.

   Using our last example:

   L	= 4.5
   nb	= 1
   nw	= 3
   i	= 1
   N	= 6

   

   Properties

   1. Not sensitive to all scores.

      Xs		Md
      1, 2, 3	2	2
      1, 2, 30	11	2
      1, 2, 300	101	2
      1, 2, 3000	1001	2

      
      

   2. Most useful with skewed distributions.
      

3. Mode
   Is the most frequently occurring score. Note:
   • There can be more than one. Can have bi- or tri-modal distributions and then speak of major and minor modes.
   • It is symbolized as M_o.
   • For grouped data, we have a Modal Interval and a Crude M_o.
     Considering the example discussed in the section on Frequency Distributions, the Modal Interval is 80-84 and the Crude M_o is 82.
     

Note that the presence and direction of skew in the distribution can be determined from the mean and median. The key to understanding this is to be aware that the mean is sensitive to all scores, while the median is not. There are three rules:

1. If - M_d > 0 then +skew
   
2. If - M_d < 0 then -skew
   
3. If - M_d = 0 then the distribution is normal
   and all three measures of central tendency coincide.
   
   

Variability refers to the extent to which the scores in a distribution differ from each other. An equivalent definition (that is easier to work with mathematically) says that variability refers to the extent to which the scores in a distribution differ from their mean. If a distribution is lacking in variability, we may say that it is homogenous (note the opposite would be heterogenous).

We will discuss four measures of variability for now: the range, mean or average deviation, variance and standard deviation.

1. Range
   As we noted when discussing the rules for creation of a grouped frequency distribution, the range is given by the highest score in the distribution minus the lowest score plus one.
   
   R = X_H - X_L+ 1
   Example:
   
   

   
   
   Distribution A has a larger range (and more variability) than Distribution B.
   
   

   Because only the two extreme scores are used in computing the range, however, it is a crude measure. For example:
   
   

   
   
   The range of Distribution A and B is the same, although Distribution A has more variability.

2. Mean (or Average) Deviation
   If a deviation (c) is the difference of a score from its mean and variability is the extent to which the scores differ from their mean, then summing all the deviations and dividing by the number of them should give us a measure of variability. The problem though is that the deviations sum to zero. However, computing the absolute value of the deviations before summing them eliminates this problem. Thus, the formula for the MD is given by:

   

   The problem with the MD is that due to the use of the absolute value, it is a terminal procedure. In other words, it cannot be used in further calculations (which is something that we would like to be able to do).

3. Variance
   Another solution to the problem of the deviations summing to zero is to square the deviations. That is:
   
   
   Thus another name for the Variance is the Mean of the Squared Deviations About the Mean (or more simply, the Mean of Squares (MS)). The problem with the MS is that its units are squared and thus represent space, rather than a distance on the X axis like the other measures of variability.

4. Standard Deviation
   A simple solution to the problem of the MS representing a space is to compute its square root. That is:
   

   

   Since the stardard deviation can be very small, it is usually reported with 2-3 more decimals of accuracy that what is available in the original data.

   Properties of the Variance & Standard Deviation:

   1. Are always positive (or zero).
   2. Equal zero when all scores are identical (i.e., there is no variability).
   3. Like the mean, they are sensitive to all scores.

Estimation is the goal of inferential statistics. We use sample values to estimate population values. The symbols are as follows:

Measure	Sample	Population
Mean
Variance	s2	s2
Standard Deviation	s	s

It is important that the sample values (estimators) be unbiased. An unbiased estimator of a parameter is one whose average over all possible random samples of a given size equals the value of the parameter.

While is an unbiased estimator of m, s² is not an unbiased estimator of s².

In order to make it an unbiased estimator, we use N-1 in the denominator of the formula rather than just N. Thus:

Note that this is a defining formula and, as we will see below, is not the best choice when actually doing the calculations.

Let's reconsider an example from above of two distributions (A & B):

Consider a possibility for the scores that go with these distributions:

Distribution	A	B
Data	150	150
	145	110
	100	100
	100	100
	55	90
	50	50
å	600	600
N	6	6
	100	100
Range	150-50+1=101	150-50+1=101

Notice that the central tendency and range of the two distributions are the same. That is, the mean, median, and mode all equal 100 for both distributions and the range is 101 for both distributions. However, while Distributions A and B have the same measures of central tendency and the same range, they differ in their variability. Distribution A has more of it. Let us prove this by computing the standard deviation in each case. First, for Distribution A:

	A		c	c2
	150	100	50	2500
	145	100	45	2025
	100	100	0	0
	100	100	0	0
	55	100	-45	2025
	50	100	-50	2500
å	600		0	9050
N	6

Plugging the appropriate values into the defining formula gives:

Measure	A

Note that calculating the variance and standard deviation in this manner requires computing the mean and subtracting it from each score. Since this is not very efficient and can be less accurate as a result of rounding error, a computational formula is typically used. It is given as follows:

Redoing the computations for Distribution A in this manner gives:

	A	X2
	150	22500
	145	21025
	100	10000
	100	10000
	55	3025
	50	2500
å	600	69050
N	6

Then, plugging in the appropriate values into the computational formula gives:

Note that the defining and computational formulas give the same result, but the computational formula is easier to work with (and potentially more accurate due to less rounding error).

Doing the same calculations for Distribution B yields:

	B	X2
	150	22500
	110	12100
	100	10000
	100	10000
	90	8100
	50	2500
å	600	65200
N	6

Then, plugging in the appropriate values into the computational formula gives:

Thus, Distribution A clearly has more variability than Distribution B.