Method for calculating the mean
Method for calculating the mean
Step 1. Add all of the observed values in the distribution.
Step 2. Divide the sum by the number of observations.
EXAMPLE: Finding the Mean
Find the mean of the following incubation periods for hepatitis A: 27, 31, 15, 30, and 22 days.
Step 1. Add all of the observed values in the distribution.
Step 2. Divide the sum by the number of observations.
Therefore, the mean incubation period is 25.0 days.
Properties and uses of the arithmetic mean
• The mean has excellent statistical properties and is commonly used in additional statistical manipulations and analyses. One
To identify the mean from
such property is called the centering property of the mean.
a data set in Analysis Module:
When the mean is subtracted from each observation in the data set, the sum of these differences is zero (i.e., the negative sum is
Click on the Means
equal to the positive sum). For the data in the previous hepatitis
command under the
A example:
Statistics folder In the Means Of drop-
Value minus Mean
Difference
down box, select the variable of interest
25.0 -10.0 Select Variable
25.0 -3.0 Click OK
You should see the
list of the frequency by
the variable you
13.0 = 0 selected. Scroll down
until you see the Mean
among other data.
Mean: the center of
This demonstrates that the mean is the arithmetic center of the
gravity of the distribution
distribution.
• Because of this centering property, the mean is sometimes called the center of gravity of a frequency distribution. If the
frequency distribution is plotted on a graph, and the graph is balanced on a fulcrum, the point at which the distribution would
balance would be the mean.
• The arithmetic mean is the best descriptive measure for data that
are normally distributed.
• On the other hand, the mean is not the measure of choice for data that are severely skewed or have extreme values in one
direction or another. Because the arithmetic mean uses all of the observations in the distribution, it is affected by any extreme value. Suppose that the last value in the previous distribution was 131 instead of 31. The mean would be 225 / 5 = 45.0 rather than 25.0. As a result of one extremely large value, the mean is much larger than all values in the distribution except the extreme value (the “outlier”).
Epi Info Demonstration: Finding the Median
Question:
In the data set named SMOKE, what is the mean weight of the participants?
Answer:
In Epi Info: Select Analyze Data.
Select Read (Import). The default data set should be Sample.mdb. Under Views, scroll down to view SMOKE, and double click, or click once and then click OK. Note that 9 persons have a weight of 777, and 10 persons have a weight of 999. These are code for “refused” and “missing.” To delete these records, enter the following commands:
Click on Select. Then type in the weight < 770, or select weight from available values, then type
< 750, and click on OK. Select Means. Then click on the down arrow beneath Means of, scroll down and select WEIGHT, then click OK.
The resulting output should indicate a mean weight of 158.116 pounds.
Your Turn:
What is the mean number of cigarettes smoked per day? [Answer: 17]
Exercise 2.5
Determine the mean for the same set of vaccination data.
Check your answers on page 2-60
The midrange (midpoint of an interval) Definition of midrange
The midrange is the half-way point or the midpoint of a set of observations. The midrange is usually calculated as an intermediate step in determining other measures.
Method for identifying the midrange
Step 1. Identify the smallest (minimum) observation and the largest (maximum) observation.
Step 2. Add the minimum plus the maximum, then divide by two.
Exception: Age differs from most other variables because age does not follow the usual rules for rounding to the nearest integer. Someone who is 17 years and 360 days old cannot claim to be 18 year old for at least 5 more days. Thus, to identify the midrange for age (in years) data, you must add the smallest (minimum) observation plus the largest (maximum) observation plus 1, then divide by two.
Midrange (most types of data) = (minimum + maximum) / 2 Midrange (age data) = (minimum + maximum + 1) / 2
Consider the following example:
In a particular pre-school, children are assigned to rooms on the basis of age on September 1. Room 2 holds all of the children who were at least 2 years old but not yet 3 years old as of September 1. In other words, every child in room 2 was 2 years old on September 1. What is the midrange of ages of the children in room
2 on September 1?
For descriptive purposes, a reasonable answer is 2. However, recall that the midrange is usually calculated as an intermediate step in other calculations. Therefore, more precision is necessary.
Consider that children born in August have just turned 2 years old. Others, born in September the previous year, are almost but not quite 3 years old. Ignoring seasonal trends in births and assuming a very large room of children, birthdays are expected to be uniformly distributed throughout the year. The youngest child, born on September 1, is exactly 2.000 years old. The oldest child, whose birthday is September 2 of the previous year, is 2.997 years old. For statistical purposes, the mean and midrange of this theoretical group of 2-year-olds are both 2.5 years.