# Lesson 2: Summarizing Data

## Section 6: Measures of Central Location

A measure of central location provides a single value that summarizes an entire distribution of data. Suppose you had data from an outbreak of gastroenteritis affecting 41 persons who had recently attended a wedding. If your supervisor asked you to describe the ages of the affected persons, you could simply list the ages of each person. Alternatively, your supervisor might prefer one summary number — a measure of central location. Saying that the mean (or average) age was 48 years rather than reciting 41 ages is certainly more efficient, and most likely more meaningful.

Measures of central location include the mode, median, arithmetic mean, midrange, and geometric mean. Selecting the best measure to use for a given distribution depends largely on two factors:

• The shape or skewness of the distribution, and
• The intended use of the measure.

Each measure — what it is, how to calculate it, and when best to use it — is described in this section.

### Mode

#### Definition of mode

The mode is the value that occurs most often in a set of data. It can be determined simply by tallying the number of times each value occurs. Consider, for example, the number of doses of diphtheria-pertussis-tetanus (DPT) vaccine each of seventeen 2-year-old children in a particular village received:

0, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4

Two children received no doses; two children received 1 dose; three received 2 doses; six received 3 doses; and four received all 4 doses. Therefore, the mode is 3 doses, because more children received 3 doses than any other number of doses.

#### Method for identifying the mode

1. Step 1. Arrange the observations into a frequency distribution, indicating the values of the variable and the frequency with which each value occurs. (Alternatively, for a data set with only a few values, arrange the actual values in ascending order, as was done with the DPT vaccine doses above.)
2. Step 2. Identify the value that occurs most often.

#### EXAMPLES: Identifying the Mode

Example A: Table 2.8 (below) provides data from 30 patients who were hospitalized and received antibiotics. For the variable "length of stay" (LOS) in the hospital, identify the mode.

1. Step 1. Arrange the data in a frequency distribution.
LOS
Frequency
0
1
1
0
2
1
3
1
4
1
5
2
6
1
7
1
8
1
9
3
LOS
Frequency
10
5
11
1
12
3
13
1
14
1
15
0
16
1
17
0
18
2
19
1
LOS
Frequency
20
0
21
0
22
1
.
0
.
0
27
1
.
0
.
0
49
1
Alternatively, arrange the values in ascending order.
0, 2, 3, 4, 5, 5, 6, 7, 8, 9,
9, 9, 10, 10, 10, 10, 10, 11, 12, 12,
12, 13, 14, 16, 18, 18, 19, 22, 27, 49
2. Step 2. Identify the value that occurs most often.
Most values appear once, but the distribution includes two 5s, three 9s, five 10s, three 12s, and two 18s.
Because 10 appears most frequently, the mode is 10.

Example B: Find the mode of the following incubation periods for hepatitis A: 27, 31, 15, 30, and 22 days.

1. Step 1. Arrange the values in ascending order.
15, 22, 27, 30, and 31 days
2. Step 2. Identify the value that occurs most often.
None

Note: When no value occurs more than once, the distribution is said to have no mode.

Example : Find the mode of the following incubation periods for Bacillus cereus food poisoning:

2, 3, 3, 3, 3, 3, 4, 4, 5, 6, 7, 9, 10, 11, 11, 12, 12, 12, 12, 12, 14, 14, 15, 17, 18, 20, 21 hours
1. Step 1. Arrange the values in ascending order.
Done
2. Step 2. Identify the values that occur most often.
Five 3s and five 12s

Example C illustrates the fact that a frequency distribution can have more than one mode. When this occurs, the distribution is said to be bi-modal. Indeed, Bacillus cereus is known to cause two syndromes with different incubation periods: a short-incubation- period (1–6 hours) syndrome characterized by vomiting; and a long-incubation-period (6–24 hours) syndrome characterized by diarrhea.

Table 2.8 Sample Data from the Northeast Consortium Vancomycin Quality Improvement Project

ID Admission Date Discharge Date LOS DOB (mm/dd) DOB (year) Age Sex ESRD
• Page last reviewed: May 18, 2012
• Page last updated: May 18, 2012
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