MEAN

The mean is the average of a group of numbers. Its symbol is X¯ in statistical analysis. We sometimes use the mean value to perform other statistical tests on the group. The mean is one indicator of “central tendency.” Central tendency is the distribution of data around a central value.

Example 13–1

A student was performing a manual wet chemistry laboratory in which she obtained five replicate absorbance values for her unknown. What is the mean absorbance value for the unknown?

Replicate	Absorbance Value
1	0.425
2	0.430
3	0.435
4	0.432
5	0.428

To calculate the mean, first add the five absorbance values.

0.425+0.430+0.435+0.432+0.428=2.15

The next step is to divide the sum by the total number of absorbances, in this case 5. The total quantity of numbers in a set of numbers is referred to as “n.”

The mean value for this group of absorbance values is 0.430.

In this case, the mean value happened to also be the same as one of the numbers of the group. This does not happen all of the time.

Additional Examples

 What is the mean of the following group of magnesium values?

    3.5 mg/dL, 5.4 mg/dL, 2.8 mg/dL, 4.1 mg/dL, 2.5 mg/dL, 3.6 mg/dL, 4.7 mg/dL

The mean value for this group of numbers is 3.8 mg/dL.

 What is the mean of the following group of sodium values?

    144 mEq/L, 137 mEq/L, 141 mEq/L, 143 mEq/L, 138 mEq/L, 140 mEq/L, 141 mEq/L, 146 mEq/L, 147 mEq/L, 140 mEq/L

The mean value for this group of numbers is 142 mEq/L.

MEDIAN

Another indicator of central tendency is the median. The median is the central number in a group of numbers when the numbers are arranged in sequential order. In a group of numbers, an equal amount of the numbers is greater than the median value and an equal amount of the numbers is less than the median value. The median value may or may not be the same as the mean value.

Example 13–2

Using the same group of absorbance values as in Example 13–1, find the median value.

To find the median value, first rank the numbers from lowest to highest value.

Lowest value	0.425
	0.428
	0.430
	0.432
Highest value	0.435

Notice that the total number of values (5) in this set of values is an odd number. To determine the median of a set of values that contain an odd number of values, first add 1 to the total number of values, in this case 5. Divide the total number of values (plus 1) by 2.

In this example, there are five values.

5+1=662=3

Next, go down the list to the third number on the list, which is 0.430. This number is the median value for this group of numbers.

Lowest value	0.425
	0.428
	0.430	←MEDIAN
	0.432
Highest value	0.435

Notice that there is an equal quantity of numbers above and below the median value.

Additional Examples

 Find the median of the following group of creatinine values:

1.2 mg/dL, 1.0 mg/dL, 1.3 mg/dL, 1.1 mg/dL, 1.4 mg/dL, 1.2 mg/dL, 1.1 mg/dL

    To find the median value, first rank the group of numbers from lowest value to highest value: 1.0, 1.1, 1.1, 1.2, 1.2, 1.3, 1.4.

    Since there are 7 numbers, add 1 to the number 7, divide by 2 which is 4, and find the fourth number: 1.2. Therefore the median value in this group of creatinine values is 1.2 mg/dL.

 Find the median value in the following group of glucose values:

88 mg/dL, 75 mg/dL, 79 mg/dL, 81 mg/dL, 83 mg/dL

    Since there are five values, after ranking the numbers from lowest to highest value, find the third value in the list of numbers [(5 +1)/2 = 3]

    Therefore the median value in this group of glucose values is 81 mg/dL.

Example 13–3

In Example 13–2, the quantity of numbers in the group was an odd number: 5. If the total of a group of numbers is an even number, the median must be calculated slightly differently. The following is a group of numbers. What is the median value?

27,24,22,25,20,21,23,26

The first step is to arrange the numbers in order from lowest to highest value.

Lowest	20
	21
	22
	23
	24
	25
	26
Highest	27

Next, divide the total quantity of numbers in this group by 2. There are eight numbers in this group.

82=4

Next, add 1 to this value.

4+1=5

The median in this group of numbers is found to be the average of the fourth and fifth numbers in this group.

20
21
22
23	Fourth number = Median = 23.5
24	Fifth number
25
26
27

The median for this group of numbers is 23.5. Notice that there is an equal quantity of numbers above and below the median value.

Additional Examples

 What is the median value for this set of chloride values?

107 mEq/L, 105 mEq/L, 110 mEq/L, 109 mEq/L

To calculate the median, first rank from lowest to highest value:

105, 107, 109, 110

Next, since there are four values, divide 4 by 2, yielding 2; then add 1, equaling 3. Therefore the median value will be the average of the second and third value, or 108 mEq/L.

 What is the median value for this set of BUN values?

10.8mg/dL,10.2mg/dL,10.7mg/dL,10.2mg/dL,10.3mg/dL,10.4mg/dL

First, rank from lowest to highest value:

10.2,10.2,10.3,10.4,10.7,10.8

Next, take the average of the third and fourth value (6 values/2 = 3, + 1 = 4). Therefore, the median for this group of BUN values is 10.35, rounded to 10.4 mg/dL.

MODE

The third indicator of central tendency is the mode. The mode is the number that occurs most frequently in a group of numbers.

Example 13–4

What is the modal number for the following group of glucose values? All values are in milligrams/deciliter units.

75,74,72,70,76,73,72,71

The first step is to arrange the data from the lowest value to the highest.

Lowest	70
	71
	72
	72
	73
	74
	75
Highest	76

Next, determine the number or numbers that occur the most frequently.

Number	Frequency
70	1
71	1
72	1
73	1
74	1
75	1
76	1

From this chart, you can see that the number 72 occurs two times. 72 mg/dL is the modal number for this group of glucose values.

Additional Examples

 Calculate the modal value of the following group of potassium values:

4.5mEq/L,4.6mEq/L,4.5mEq/L,4.7mEq/L

To solve, rank the values from lowest to highest value:

4.5,4.5,4.6,4.7

The modal value is 4.5 mEq/L.

 Calculate the modal value of the following group of AST values:

36IU/L,22IU/L,25IU/L,21IU/L,22IU/L

The modal value is 22 IU/L.

GAUSSIAN DISTRIBUTION

Mean, median, and mode are all indicators of central tendency. When the mean, median, and mode are all the same number, we say that the group of numbers has a gaussian distribution. Figure 13–1 is a gaussian distribution. Another term for gaussian distribution is normal distribution. A gaussian distribution is bell-shaped with an equal amount of results above and below the highest point of the bell. In the laboratory, many times it is assumed that our data have a gaussian distribution when we perform statistical analysis of the data. It is outside of the scope of this book to go into detail about the statistical analysis of data that do not have a gaussian distribution. In a gaussian distribution, as the mean, median, and mode are equal, a frequency distribution of the data has a bell shape.

ACCURACY VERSUS PRECISION

In the laboratory, we strive to produce results that correctly assess the patient’s condition. Two terms are used to describe the quality of the results that are produced. These are accuracy and precision. An accurate result is a result that correctly reflects the “true value” of the result. Imagine a “target” value that should be obtained for an analyte. If the results that are obtained are of the target value, then we can say that the results are accurate. The mean value we obtained in our data should be very close to the target value. In the laboratory, by using quality assurance, we try to ensure that the tendency for the results to be centrally located or “hitting the target” can be achieved.

Precision occurs when, after repeated analysis, the same result is achieved. Precision is related to the amount of dispersion of the data around the target value. If there is a wide degree of dispersion, the results are not precise. However, precise results are those that are tightly clustered together. In the laboratory, we want results that are tightly clustered around the true value: that is, both accurate and precise. The statistical tools of variance, standard deviation, and coefficient of variation help assess the degree of dispersion of the data.

VARIANCE

The precision of a group of numbers is indicated by the variance. The symbol for variance is “s².” The variance indicates how close together, or how precise, are the numbers within a group. A group of numbers with a large variance would be expected to have a wide range of values. A group of numbers with a small variance would be expected to have numbers that are very close in value. The smaller the variance of a group of numbers, the more precise they are. Variance is calculated using the following formula:

s2=∑(Xd−X¯)2n−1

where:

s² = variance

∑ = the sum of the numbers within the parentheses

X_d = an individual data point within the group

X¯ = the mean of the group of numbers

n = the total amount of numbers within the group

Most scientific calculators have a variance function. By using the statistical mode of the calculator, the variance can be easily obtained. The variance can also be calculated using computer software such as Microsoft Excel. Manually calculating the variance of a group of numbers may be time-consuming, but may be necessary if a scientific calculator or computer is not available.

To calculate the variance of a group of numbers, the mean of the group is first determined. Then the mean is subtracted from each individual number in the group. Sometimes a negative number will result if the mean is of a greater value than a particular number in the group. The differences will be individually squared. The next step is to determine the sum of the individually squared numbers. This value is the numerator of the formula and is divided by one less than the total amount of numbers in the group (n). The number obtained from this calculation is the variance of the group. The units associated with the variance value will also be squared and will not be the same as the units of the group of numbers for which the variance was calculated.

Example 13–5

Calculate the variance of the following group of glucose values. All values have the unit of milligrams per deciliter:

94,93,92,100,101,93,105,98,99,87

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