Of the following list, which role would a human service professional not assume?
Section 6: Measures of Central LocationA 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. Show
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:
Each measure — what it is, how to calculate it, and when best to use it — is described in this section. ModeDefinition of modeThe 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
EXAMPLES: Identifying the ModeExample 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.
Example B: Find the mode of the following incubation periods for hepatitis A: 27, 31, 15, 30, and 22 days.
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
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
To identify the mode from a data set in Analysis Module: Epi Info does not have a Mode command. Thus, the best way to identify the mode is to create a histogram and look for the tallest column(s). Select graphs, then choose histogram under Graph Type. The tallest column(s) is(are) the mode(s). NOTE: The Means command provides a mode, but only the lowest value if a distribution has more than one mode. Properties and uses of the modeThe mode is the easiest measure of central location to understand and explain. It is also the easiest to identify, and requires no calculations.
Exercise 2.3Using the same vaccination data as in Exercise 2.2, find the mode. (If you answered Exercise 2.2, find the mode from your frequency distribution.) 2, 0, 3, 1, 0, 1, 2, 2, 4, 8, 1, 3, 3, 12, 1, 6, 2, 5, 1 Check your answers MedianDefinition of medianThe median is the middle value of a set of data that has been put into rank order. Similar to the median on a highway that divides the road in two, the statistical median is the value that divides the data into two halves, with one half of the observations being smaller than the median value and the other half being larger. The median is also the 50th percentile of the distribution. Suppose you had the following ages in years for patients with a particular illness: 4, 23, 28, 31, 32 The median age is 28 years, because it is the middle value, with two values smaller than 28 and two values larger than 28. Method for identifying the medianStep 1. Arrange the observations into increasing or decreasing order. Step 2. Find the middle position of the distribution by using the following formula: Middle position = (n + 1) / 2
Step 3. Identify the value at the middle position.
Properties and uses of the median
Exercise 2.4Determine the median for the same vaccination data used in Exercises 2.2. and 2.3. 2, 0, 3, 1, 0, 1, 2, 2, 4, 8, 1, 3, 3, 12, 1, 6, 2, 5, 1 Check your answers Arithmetic meanDefinition of meanThe arithmetic mean is a more technical name for what is more commonly called the mean or average. The arithmetic mean is the value that is closest to all the other values in a distribution. Method for calculating the meanStep 1. Add all of the observed values in the distribution. Step 2. Divide the sum by the number of observations. EXAMPLE: Finding the MeanFind 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. 27 + 31 + 15 + 30 + 22 = 125 Step 2. Divide the sum by the number of observations. 125 / 5 = 25.0 Therefore, the mean incubation period is 25.0 days. Properties and uses of the arithmetic mean
This demonstrates that the mean is the arithmetic center of the distribution.
Epi Info Demonstration: Finding the MedianQuestion: In the data set named SMOKE, what is the mean weight of the participants? Answer: In Epi Info: 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 2.5Determine the mean for the same set of vaccination data. 2, 0, 3, 1, 0, 1, 2, 2, 4, 8, 1, 3, 3, 12, 1, 6, 2, 5, 1 Check your answers The midrange (midpoint of an interval)Definition of midrange Method for identifying the midrange
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 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. Properties and uses of the midrange
EXAMPLES: Identifying the MidrangeExample A: Find the midrange of the following incubation periods for hepatitis A: 27, 31, 15, 30, and 22 days.
Example B: Find the midrange of the grouping 15–24 (e.g., number of alcoholic beverages consumed in one week).
This calculation assumes that the grouping 15–24 really covers 14.50–24.49…. Since the midrange of 14.50–24.49… = 19.49…, the midrange can be reported as 19.5. Example C: Find the midrange of the age group 15–24 years.
Midrange = (15 + 24 + 1) / 2 = 40 / 2 = 20 years Age differs from the majority of other variables because age does not follow the usual rules for rounding to the nearest integer. For most variables, 15.99 can be rounded to 16. However, an adolescent who is 15 years and 360 days old cannot claim to be 16 years old (and hence get his driver’s license or learner’s permit) for at least 5 more days. Thus, the interval of 15–24 years really spans 15.0–24.99… years. The midrange of 15.0 and 24.99… = 19.99… = 20.0 years. Geometric meanTo calculate the geometric mean, you need a scientific calculator with log and yx keys. Definition of geometric mean More About LogarithmsA logarithm is the power to which a base is raised. To what power would you need to raise a base of 10 to
get a value of 100? 20 = 1 (anything raised to the 0 power is 1) 100 = 1 (Anything raised to the 0 power equals 1) An antilog raises the base to the power (logarithm). For example, the antilog of 2 at base 10 is 102, or 100. The antilog of 4 at base 2 is 24, or 16. The majority of titers are reported as multiples of 2 (e.g., 2, 4, 8, etc.); therefore, base 2 is typically used when dealing with titers. Method for calculating the geometric meanThere are two methods for calculating the geometric mean. Method A
Method B
EXAMPLES: Calculating the Geometric Mean Example A: Using Method A 10, 10, 100, 100, 100, 100, 10,000, 100,000, 100,000, 1,000,000 Because these values are all multiples of 10, it makes sense to use logs of base 10. Take the log (in this case, to base 10) of each value. log10(xi) = 1, 1, 2, 2, 2, 2, 4, 5, 5, 6 Calculate the mean of the log values by summing and dividing by the number of observations (in this case, 10). Mean of log10(xi) = (1+1+2+2+2+2+4+5+5+6) / 10 = 30 / 10 = 3
Example B: Using Method B Calculate the geometric mean from the following 95% confidence intervals of an odds ratio: 1.0, 9.0
1.0 x 9.0 = 9.0
The geometric mean = square root of 9.0 = 3.0. Properties and uses of the geometric meanThe geometric mean is the average of logarithmic values, converted back to the base. The geometric mean tends to dampen the effect of extreme values and is always smaller than the corresponding arithmetic mean. In that sense, the geometric mean is less sensitive than the arithmetic mean to one or a few extreme values.
Exercise 2.6Using the dilution titers shown below, calculate the geometric mean titer of convalescent antibodies against tularemia among 10 residents of Martha’s Vineyard. [Hint: Use only the second number in the ratio, i.e., for 1:640, use 640.]
Check your answers Selecting the appropriate measureMeasures of central location are single values that summarize the observed values of a distribution. The mode provides the most common value, the median provides the central value, the arithmetic mean provides the average value, the midrange provides the midpoint value, and the geometric mean provides the logarithmic average. The mode and median are useful as descriptive measures. However, they are not often used for further statistical manipulations. In contrast, the mean is not only a good descriptive measure, but it also has good statistical properties. The mean is used most often in additional statistical manipulations. While the arithmetic mean is the measure of choice when data are normally distributed, the median is the measure of choice for data that are not normally distributed. Because epidemiologic data tend not to be normally distributed (incubation periods, doses, ages of patients), the median is often preferred. The geometric mean is used most commonly with laboratory data, particularly dilution titers or assays and environmental sampling data. The arithmetic mean uses all the data, which makes it sensitive to outliers. Although the geometric mean also uses all the data, it is not as sensitive to outliers as the arithmetic mean. The midrange, which is based on the minimum and maximum values, is more sensitive to outliers than any other measures. The mode and median tend not to be affected by outliers. In summary, each measure of central location — mode, median, mean, midrange, and geometric mean — is a single value that is used to represent all of the observed values of a distribution. Each measure has its advantages and limitations. The selection of the most appropriate measure requires judgment based on the characteristics of the data (e.g., normally distributed or skewed, with or without outliers, arithmetic or log scale) and the reason for calculating the measure (e.g., for descriptive or analytic purposes). Exercise 2.7For each of the variables listed below from the line listing in Table 2.9, identify which measure of central location is best for representing the data.
________ 6. Year of diagnosis ________ 7. Age (years) ________ 8. Sex ________ 9. Highest IFA titer ________ 10. Platelets x 106/L ________ 11. White blood cell count x 109/L Table 2.9 Line Listing for 12 Patients with Human Monocytotropic Ehrlichiosis — Missouri, 1998–1999
*Immunofluorescence assay Data Source: Olano JP, Masters E, Hogrefe W, Walker DH. Human monocytotropic ehrlichiosis, Missouri. Emerg Infect Dis 2003;9:1579-86. Check your answers Top of Page What is human service quizlet?Human Services. To assist individual and communities to function as effectively as possible in the major domains of living. Human Service Networks. Programs and entitlements that offer help in dealing with different but complementary parts of an overall problem.
Which association is mainly for human service professionals?The National Organization for Human Services
NOHS is a membership organization that works to transform individuals and communities. They do this by providing professional development opportunities, promoting certifications, and advocating for social change.
Which of the following is the best definition of homeostasis Human Services?Homeostasis is the ability to maintain internal stability in an organism in response to the environmental changes. The internal temperature of the human body is the best example of homeostasis.
Who believed that introspection and reflection?Wundt's Psychological Research
Wundt believed that there were two key components that make up the contents of the human mind: sensations and feelings. Wundt focused on making the introspection process as structured and precise as possible.
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