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Answers
The distinction between descriptive and inferential statistics is fundamental, and a set of notational conventions and terminology has been developed to distinguish between the two. Although these conventions differ somewhat from one author to the next, as a general rule, numbers that describe a population are referred to as parameters and are signified by Greek letters such as µ (for the population mean) and σ (for the population standard deviation); numbers that describe a sample are referred to as statistics and are signified by Latin letters such as (the sample mean) and s (the sample standard deviation).
Measures of Central Tendency
Measures of central tendency, also known as measures of location, are typically among the first statistics computed for the continuous variables in a new data set. The main purpose of computing measures of central tendency is to give you an idea of what a typical or common value for a given variable is. The three most common measures of central tendency are the arithmetic mean, the median, and the mode.
The Mean
The arithmetic mean, or simply the mean, is often referred to in ordinary speech as the average of a set of values. Calculating the mean as a measure of central tendency is appropriate for interval and ratio data, and the mean of dichotomous variables coded as 0 or 1 provides the proportion of subjects whose value on the variable is 1. For continuous data, for instance measures of height or scores on an IQ test, the mean is simply calculated by adding up all the values and then dividing by the number of values. The mean of a population is denoted by the Greek letter mu (µ) whereas the mean of a sample is typically denoted by a bar over the variable symbol: for instance, the mean of x would be written and pronounced “x-bar.” Some authors adapt the bar notation for the names of variables also. For instance, some authors denote “the mean of the variable age” by , which would be pronounced “age-bar.”
Suppose we have a population with only five cases, and these are the values for members of that population for the variable x:
100, 115, 93, 102, 97
We can calculate the mean of x by adding these values and dividing by 5 (the number of values):
µ = (100 + 115 + 93 + 102 + 97)/5 = 507/5 = 101.4
Statisticians often use a convention called summation notation, introduced in Chapter 1, which defines a statistic by describing how it is calculated. The computation of the mean is the same whether the numbers are considered to represent a population or a sample; the only difference is the symbol for the mean itself. The mean of a population, as expressed in summation notation.