8a.4 Measures of Dispersion

Chapter 8a: Descriptive Statistics 8a.1 Introduction
8a.2 Kinds of Measurements
8a.3 Measures of Central Tendency
8a.4 Measures of Dispersion
8a.5 Displaying Data

8a.4 Measures of Dispersion

The previous section presented the mean, median, and mode as measures of central tendency. These are important values, but they only describe the "center" of a set of data. For example, each of the seven data sets given below has a mean of 100, a median of 110, and a mode of 105.

Label	Data Set
A	{72, 72, 73, 73, 105, 105, 105, 110, 110, 111, 111, 112, 112, 113, 116}
B	{5, 5, 6, 105, 105, 105, 109, 110, 125, 125, 130, 135, 140, 145, 150}
C	{-710, -706, -704, -700, 80, 105, 105, 110, 557, 558, 559, 560, 561, 562, 563}
D	{-2510, -106, -105, -100, 80, 105, 105, 110, 207, 208, 209, 210, 211, 212, 2664}
E	{80, 81.6, 81.7, 81.9, 82, 105, 105, 110, 110.1, 110.2, 110.3, 110.4, 110.5, 110.6, 110.7}
F	{-710, 80, 94, 96, 100, 105, 105, 110, 157, 158, 159, 160, 161, 162, 563}
G	{-710, 60, 74, 76, 100, 105, 105, 110, 111, 112, 113, 114, 222, 345, 563}

The mean, median, and mode values of these seven data sets are fixed, and yet the data values in the sets are clearly different. Measures of central tendency characterize the "center" of the data but they do not give us any information about the "spread" of that data. Characterizing the "spread" of the data values is the task of the measures of dispersion, the range, the quartile points, and the standard deviation. The following table gives the values for these measures for each of the data sets given above.

Label	Data Set	Range	First Quartile	Second Quartile	Third Quartile	Standard Deviation
A	{72, 72, 73, 73, 105, 105, 105, 110, 110, 111, 111, 112, 112, 113, 116}	72 to 116	73	110	112	16.844
B	{5, 5, 6, 105, 105, 105, 109, 110, 125, 125, 130, 135, 140, 145, 150}	5 to 150	105	110	135	49.425
C	{-710, -706, -704, -700, 80, 105, 105, 110, 557, 558, 559, 560, 561, 562, 563}	-710 to 563	-700	110	560	521.148
D	{-2510, -106, -105, -100, 80, 105, 105, 110, 207, 208, 209, 210, 211, 212, 2664}	-2510 to 2664	-100	110	210	951.601
E	{80, 81.6, 81.7, 81.9, 82, 105, 105, 110, 110.1, 110.2, 110.3, 110.4, 110.5, 110.6, 110.7}	80 to 110.7	81.9	110	110.4	13.248
F	{-710, 80, 94, 96, 100, 105, 105, 110, 157, 158, 159, 160, 161, 162, 563}	-710 to 563	96	110	160	243.903
G	{-710, 60, 74, 76, 100, 105, 105, 110, 111, 112, 113, 114, 222, 345, 563}	-710 to 563	76	110	114	251.530

The Range

The range is the smallest and largest value in the data set, that is, the minimum and maximum values. The range is applicable to ordinal, interval, and ratio measurements. Nominal measurements are merely names for values. We cannot use the order of the numbers that we assign as names as a process to give an order to the underlying data.

Once we know the range, we at least know the boundaries of the data. The following table reproduces the data set Labels and the range values given above.

Label	Range
A	72 to 116
B	-5 to 150
C	-710 to 563
D	-2510 to 2664
E	80 to 110.7
F	-710 to 563
G	-710 to 563

Just looking at the range values in the table, without seeing the original data sets, we now have a feel for the spread of the values in each data set. We know that all seven data sets have a mean of 100 and a median of 110. Knowing the range immediately tells us that data set D has some extreme values, whereas data sets A and E stay fairly close to the mean and the median.

Although the range gives the extreme values (the minimum and maximum) in the data set, we cannot tell if there is only one extreme value or if many of the values are spread out. In the example above, data sets C, F, and G have the identical range, namely, -710 to 563. They also have the same mean, median, and mode. Just knowing the values of the measures of central tendency and knowing the range does not distinguish between the three data sets. And yet, data sets C, F, and G are quite different. Data set C has a cluster of 4 values that are very low (around -700), a cluster of 4 values in the middle (around 105), and a cluster of 7 high values (around 560). Data set F has just one low value (-710), 13 values spread between 80 and 162, and one high value (563). And, data set G has one low value (-710), 11 middle values from 60 to 114, and then three higher values (222, 345, and 563). The range is not giving us enough information to get a feel for the differences in these data sets without looking at the actual values.

Quartile Points

When we look at the difference in the data sets C, F, and G, we are really looking at the differences in the distribution of values in the data sets. One technique for characterizing those differences is through the use of quartile points. To do this we sort the values (as we did to find the median) and then we divide the list into quarters. The data value in the sorted list at the break between the first and second quarters is the 25^th percentile point, sometimes called the first quartile point. 25% of the data values in the sorted list are less than (or equal to) this value. The data value in the sorted list at the break between the second and third quarters is the 50^th percentile point, sometimes called the second quartile point. 50% of the data values in the sorted list are less than (or equal to) this value, and, therefore, this point is the median value. And, the data value in the sorted list at the break between the third and fourth quarters is the 75^th percentile point, sometimes called the third quartile point. 75% of the data values in the sorted list are less than (or equal to) this value. Naturally, if we must sort the data set values in order to find the quartile points, then we can use quartile points for ordinal, interval, and ratio measurements. We should not use quartile points for nominal measurements.

Return to the three data sets, C, F, and G, and examine their quartile points in the table below.

Label	Data Set	25%	50%	75%
C	{-710, -706, -704, -700, 80, 105, 105, 110, 557, 558, 559, 560, 561, 562, 563}	-700	110	560
F	{-710, 80, 94, 96, 100, 105, 105, 110, 157, 158, 159, 160, 161, 162, 563}	96	110	160
G	{-710, 60, 74, 76, 100, 105, 105, 110, 111, 112, 113, 114, 222, 345, 563}	76	110	114

We see the difference in the data sets reflected in the quartile points.

It should be noted that the quartile points, the first, second, and third quartile points, form a nice bridge across the data values. We extend that bridge to the extremes of the data values by adding the range values as the zeroeth and fourth quartile points. Thus, the full table of quartile points would be:

Label	Data Set	0^th Quartile 0%	1^st Quartile 25%	2^nd Quartile 50%	3^rd Quartile 75%	4^th Quartile 100%
C	{-710, -706, -704, -700, 80, 105, 105, 110, 557, 558, 559, 560, 561, 562, 563}	-710	-700	110	560	563
F	{-710, 80, 94, 96, 100, 105, 105, 110, 157, 158, 159, 160, 161, 162, 563}	-710	96	110	160	563
G	{-710, 60, 74, 76, 100, 105, 105, 110, 111, 112, 113, 114, 222, 345, 563}	-710	76	110	114	563

The concept of quartiles works quite well. It gives us a feel for the distribution of values within the data set. It is tempting to extend the concept from 4 quarters to 10 equal parts, thus obtaining the 10%, 20%, 30%, 40%, and so on points. This would be silly for the 15 values in the example data that we are using above. It would make more sense if the data set had hundreds of values in it. The difficulty with this approach, having more and more marker points, is that it is harder and harder to look at all of the marker points and make some sense out of them. Therefore, we generally stay with quartile points.

Standard Deviation

As nice as the quartile points are, they do not resolve to a single number as did the mean and the median. A single number measure of disbursement is the standard deviation from the mean, usually just called the standard deviation. This measure reflects the "square root of the average squared distance that the values of the data set are from the mean." Certainly, if we are measuring a distance from the mean we need to be able to find the mean. As a consequence, the standard deviation is applicable to interval and ratio measurements, the kind of measurement for which we can appropriately compute the mean. Standard deviation should not be computed for nominal or for ordinal measurements.

Thus far we have merely stated that the standard deviation exists and that it is applicable to interval and ratio measurements. We have not presented a way to compute the standard deviation. The formula for the standard deviation is quite complex. It is generally given as

This is a mathematical definition for

, the symbol for standard deviation. The definition states that

is equal to the square root of a quantity. That quantity is the quotient of

and

, where

is the number of items in the data set. The numerator,

, is another mathematical expression. It represents the sum of many terms. The

tells us to "obtain the sum of every term, specified by the expression that follows the symbol, for each value of i from 1 to n". We have n values in the data set, and this represents the sum of one term for each value in the data set. The individual terms are given as

where

is the i^th value in the data set, and

is the mean of the data set. In other words, for each term in the data set we need to find the difference between the term and the mean and then we need to square that difference, and we need to find the sum of all those squared differences. That sum of the squared differences is the value represented by

. We divide that sum by the number of items,

, amd we have the "mean squared differences". Then we find the square root of that quotient, giving us the "root mean squared difference". That final answer is the standard deviation.

The following table presents the values used in the calculation of the standard deviation for data set A as given above.

i	x_i		x_i-x	(x_i-x)²
1	72		-28	784
2	72		-28	784
3	73		-27	729
4	73		-27	729
5	105		5	25
6	105		5	25
7	105		5	25
8	110		10	100
9	110		10	100
10	111		11	121
11	111		11	121
12	112		12	144
13	112		12	144
14	113		13	169
15	116		16	256

sum=	1500			4256
sum/15=	100			283.7333

		square root=		16.84439

This may seem to be a messy, complex, solution, and it is. Fortunately, there is another formula for calculating the standard deviation. This is an equivalent formula, but unlike the one given above, the new formula does not require you to find the mean before you complete the other actions. The alternative formula is

In this formula, both of the

symbols tell us to add all of the terms the follow where the value of i goes from 1 through n. The first instance is

, which represents the sum of the squares of each of the data values. The second instance is

, which represents finding the sum of all the data values and then squaring that sum. This formula is easier to compute because as we go through the data values we need only compute the sum of the data values and the sum of the squares of the data values. Once that is done, then we can compute the standard deviation as given in the second version. Below is a table of the data values from set A and their squares, along with the sum of each.

i	x_i		x_i²
1	72		5184
2	72		5184
3	73		5329
4	73		5329
5	105		11025
6	105		11025
7	105		11025
8	110		12100
9	110		12100
10	111		12321
11	111		12321
12	112		12544
13	112		12544
14	113		12769
15	116		13456

sum of x_i=	1500	sum of x_i²=	154256

Thus, the value of

is 1500²=2250000, and that value divided by 15 is 150000. The sum of the squared values,

, is 154256, and the difference between 154256 and 150000 is 4256. Then we divide that value by 15 to get 283.73333, to which we apply the square root operation to get a final answer of 16.844, as we computed using the first formula.

As fortunate as it is to have the alternative formula, even more fortunate is the fact that most scientific and all graphing calculators do all of this work for us. The following table gives the screen images and an explanation for obtaining the standard deviation, and other values, on a TI-86 (a demonstration of the TI-83 follows below).

TI-86 Version: Figure 1 Figure 1 shows the statement used to create a list, called A, which contains the values in our data set. Note that the calculator started with a clear screen, , but that we opened the LIST menu via the keys , and that we used the and keys to select the { and the } from that menu. The comma was generated via the key. The "store" symbol, , was the result of the key. That key put the calculator into alphabetic mode. As a result, the A was generated by pressing the key.
TI-86 Version: Figure 2 Figure 2 is a result of pressing the key to perform the command given in Figure 1.
TI-86 Version: Figure 3 The goal of Figure 3 is to generate the statement OneVar A. To do this on the TI-86 (the TI-85 is quite different) we open the STAT menu via and then open the CALC submenu via the key. Our desired command is in the first option poisition, so we press to paste "OneVar " onto the screen. We complete the statement via the keys to generate the final A.

TI-86 Version: Figure 4 Press the key to perform the command and the TI-86 responds with Figure 4. In Figure 4 we note that the mean, , is indeed 100. This screen also indicates that the sum of the values is 1500, while the sum of the squares of the values in the list is 154256, exactly the value that we calculated above. The next line on the screen, , has information that is beyond the scope of this page. The important part of that line for us is the down arrow at the left edge of the screen. This indicates that there is more information to display. We can press the key to move down to the additional items.
(The meaning of the line is: Assuming that the data values in the list are a sample of a larger population, then, we use the data values to estimate the full population standard deviation, and that estimate is 17.4355958.)

TI-86 Version: Figure 5 In Figure 5 we have moved down the display three lines by pressing the key 3 times. In addition, we have closed the sub-menu by pressing the key. As a result, we can see more information about our data set. Most important, we now see the line that gives us the standard deviation of the data values in our set. The rest of the screen gives us the information that there are 15 items in the data set, that the minimum value is 72, and that the 1^st quartile point is 73.
TI-86 Version: Figure 6 We press the key 3 more times to arrive at Figure 6. Here we find that the median is 110, the 3^rd quartile point is 112, and the maximum is 116.

We can look at the same data on a TI-83. The results will be the same but the steps are slightly different.

TI-83 Version: Figure 1 Figure 1 shows the statement used to create a list, called L₁, which contains the values in our data set. Note that the calculator started with a clear screen, . We used the keys to generate the { and the keys to generate the } symbols. The comma was generated via the key. The "store" symbol, , was the result of the key. And we selected the standard list variable L₁ via the key sequence.

TI-83 Version: Figure 2 Figure 2 is a result of pressing the key to perform the command given in Figure 1.

TI-83 Version: Figure 3 For Figure 3 we pressed the key to open the STATISTICS menu, and we have used the key to move the highlight to the CALC option. We are interested in the first sub-option, 1-Var Stats. Therefore, all we need to do is to press the key.

TI-83 Version: Figure 4 Figure 4 shows the 1-Var Stats command after we have pressed to append the L₁ to it. This is the command that we will need to run one-variable statistics on the data that is stored in the list L₁. We press the key to perform that command.

TI-83 Version: Figure 5 Figure 5 shows the first part of the output from the previous command. In This case we see that the we note that the mean, , is indeed 100. This screen also indicates that the sum of the values is 1500, while the sum of the squares of the values in the list is 154256, exactly the value that we calculated above. The next line on the screen, , gives us information that is beyond the scope of this page. However, the subsequent line, , gives us the standard deviation of our data.
The final line of output on Figure 5, , gives the value of n as 15, meaning that we have 15 values in our data set. In addition, the "down arrow to the left of the n indicates that there is more information to be seen. We merely need to press the key to show one new line of output information, and, therefore, to lose the top line of the information. We can press the key five times to change the display to see the remaining values, as shown in Figure 6.
(The meaning of the line is: Assuming that the data values in the list are a sample of a larger population, then, we use the data values to estimate the full population standard deviation, and that estimate is 17.4355958.)

TI-83 Version: Figure 6 Figure 6 repeats the value for n, along with an "up-arrow" indicating that we could move back to the Figure 5 output by using the key. The other values on Figure 6 give the minimum value as 72, the 1^st quartile point as 73, the median as 110, the 3^rd quartile point as 112, and the maximum as 116.

For your information, the web page bstat86.htm demonstrates the use of the TI-86 to find, among other things, the standard deviation (although not of the values in our example). The web page bstat85.htm demonstrates the use of the TI-85 to find, among other things, the standard deviation (although not of the values in our example). The web page begstat3.htm demonstrates the use of the TI-83 to find, among other things, the standard deviation (although not of the values in our example).

Let us return to the three data sets, C, F, and G, and look at the standard deviation of each set.

Label	Standard Deviation
C	521.114
F	243.903
G	251.530

The larger the standard deviation, the more disbursed the values. In our case, for three data sets all with the same mean, all with the same median, all with the same mode, and all with the same range, we can see that data set C has the largest standard deviation. Further, the standard deviations for data sets F and G are close, although the standard deviation of G is slightly larger than is the standard deviation of F. This corresponds to the "spread" of the values in these three data sets. Set C has many values that are far from the mean. Data sets F and G are different, but the differences are not as dramatic as are the differences between these and set C. If we look back at the second table on this page we can compare the standard deviation of each of the 7 data sets with the "spread" of values in those data sets. The standard deviation provides a single value that is representative of the "spread", the distribution, of the values in the data set. The smaller the standard deviaton, the closer the data values are to the mean.

Having looked at these measures of dispersion, we can not turn our attention to some basic methods for "Displaying Data".

Chapter 8a: Descriptive Statistics 8a.1 Introduction
8a.2 Kinds of Measurements
8a.3 Measures of Central Tendency
8a.4 Measures of Dispersion
8a.5 Displaying Data

TI-86 Version: Figure 1	Figure 1 shows the statement used to create a list, called A, which contains the values in our data set. Note that the calculator started with a clear screen, , but that we opened the LIST menu via the keys , and that we used the and keys to select the { and the } from that menu. The comma was generated via the key. The "store" symbol, , was the result of the key. That key put the calculator into alphabetic mode. As a result, the A was generated by pressing the key.
TI-86 Version: Figure 2	Figure 2 is a result of pressing the key to perform the command given in Figure 1.
TI-86 Version: Figure 3	The goal of Figure 3 is to generate the statement OneVar A. To do this on the TI-86 (the TI-85 is quite different) we open the STAT menu via and then open the CALC submenu via the key. Our desired command is in the first option poisition, so we press to paste "OneVar " onto the screen. We complete the statement via the keys to generate the final A.
TI-86 Version: Figure 4	Press the key to perform the command and the TI-86 responds with Figure 4. In Figure 4 we note that the mean, , is indeed 100. This screen also indicates that the sum of the values is 1500, while the sum of the squares of the values in the list is 154256, exactly the value that we calculated above. The next line on the screen, , has information that is beyond the scope of this page. The important part of that line for us is the down arrow at the left edge of the screen. This indicates that there is more information to display. We can press the key to move down to the additional items. (The meaning of the line is: Assuming that the data values in the list are a sample of a larger population, then, we use the data values to estimate the full population standard deviation, and that estimate is 17.4355958.)
TI-86 Version: Figure 5	In Figure 5 we have moved down the display three lines by pressing the key 3 times. In addition, we have closed the sub-menu by pressing the key. As a result, we can see more information about our data set. Most important, we now see the line that gives us the standard deviation of the data values in our set. The rest of the screen gives us the information that there are 15 items in the data set, that the minimum value is 72, and that the 1^st quartile point is 73.
TI-86 Version: Figure 6	We press the key 3 more times to arrive at Figure 6. Here we find that the median is 110, the 3^rd quartile point is 112, and the maximum is 116.

TI-83 Version: Figure 1	Figure 1 shows the statement used to create a list, called L₁, which contains the values in our data set. Note that the calculator started with a clear screen, . We used the keys to generate the { and the keys to generate the } symbols. The comma was generated via the key. The "store" symbol, , was the result of the key. And we selected the standard list variable L₁ via the key sequence.
TI-83 Version: Figure 2	Figure 2 is a result of pressing the key to perform the command given in Figure 1.
TI-83 Version: Figure 3	For Figure 3 we pressed the key to open the STATISTICS menu, and we have used the key to move the highlight to the CALC option. We are interested in the first sub-option, 1-Var Stats. Therefore, all we need to do is to press the key.
TI-83 Version: Figure 4	Figure 4 shows the 1-Var Stats command after we have pressed to append the L₁ to it. This is the command that we will need to run one-variable statistics on the data that is stored in the list L₁. We press the key to perform that command.
TI-83 Version: Figure 5	Figure 5 shows the first part of the output from the previous command. In This case we see that the we note that the mean, , is indeed 100. This screen also indicates that the sum of the values is 1500, while the sum of the squares of the values in the list is 154256, exactly the value that we calculated above. The next line on the screen, , gives us information that is beyond the scope of this page. However, the subsequent line, , gives us the standard deviation of our data. The final line of output on Figure 5, , gives the value of n as 15, meaning that we have 15 values in our data set. In addition, the "down arrow to the left of the n indicates that there is more information to be seen. We merely need to press the key to show one new line of output information, and, therefore, to lose the top line of the information. We can press the key five times to change the display to see the remaining values, as shown in Figure 6. (The meaning of the line is: Assuming that the data values in the list are a sample of a larger population, then, we use the data values to estimate the full population standard deviation, and that estimate is 17.4355958.)
TI-83 Version: Figure 6	Figure 6 repeats the value for n, along with an "up-arrow" indicating that we could move back to the Figure 5 output by using the key. The other values on Figure 6 give the minimum value as 72, the 1^st quartile point as 73, the median as 110, the 3^rd quartile point as 112, and the maximum as 116.