Mathematics
Sample Mean
The sample mean is the average of a set of numbers calculated by adding up all the values and dividing by the total count of values. It is a measure of central tendency and is often used to estimate the population mean when only a sample is available. The sample mean is a fundamental concept in statistics and is denoted by the symbol "x-bar."
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12 Key excerpts on "Sample Mean"
eBook - PDF- Evert W. Johnson(Author)
- 2000(Publication Date)
- CRC Press(Publisher)
In other words, if a number of samples of a given size are drawn from a given population, the averages of the individual samples should be reasonably similar. This criterion has its roots in the concept that sample statistics are used as estimators of population parameters and, if such an estimator is highly erratic, estimates made with it are apt to be unsatisfactory. 7. The average should be of such a nature that it can be used in statistical operations beyond its own determination. At this stage in the text, it is difficult to explain what is meant by this statement. However, it can be said that much of the remainder of this book is concerned with statistical operations within which averages are embedded. 5.3 THE ARITHMETIC MEAN The most commonly used and best known of the averages is the arithmetic mean , also known simply as the mean or the average . It can be computed regardless of the manner in which the data are arranged, but the computational procedures are dependent on the way the data are arranged. 34 Forest Sampling Desk Reference 5.3.1 U NGROUPED D ATA In its basic form, the arithmetic mean is computed simply by summing the individual values and then dividing the sum by the count of items summed. The data may be in random (raw) order or in the form of an array. Expressed algebraically, if the entire population is involved, the expression is (5.1) where µ = arithmetic mean of the population N = the total number of variates in the population, otherwise known as the population size If the dataset is from a sample , the algebraic expression is: (5.2) where = arithmetic mean of the sample n = the number of variates in the sample, otherwise known as the sample size For example, using the cone count data in either columns 1 or 2 in Table 3.1, the arithmetic mean is = 790/23 = 34.34782609 cones per tree It should be noted that one of the variates in the data set is a zero.
eBook - PDF- Frederick Gravetter, Larry Wallnau(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
In fact, the average value of all the Sample Means is exactly equal to the value of the population mean. This fact should be intuitively reasonable; the Sample Means are expected to be close to the population mean, and they do tend to pile up around μ . The formal statement of this phenomenon is that the mean of the distribution of Sample Means always is identical to the population mean. This mean value is called the expected value of M. 200 CHAPTER 7 | Probability and Samples: The Distribution of Sample Means Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. In commonsense terms, a Sample Mean is “expected” to be near its population mean. When all of the possible Sample Means are obtained, the average value is identical to μ . The fact that the average value of M is equal to μ was first introduced in Chapter 4 (page 120) in the context of biased versus unbiased statistics. The Sample Mean is an example of an unbiased statistic, which means that on average the sample statistic produces a value that is exactly equal to the corresponding population parameter. In this case, the average value of all the Sample Means is exactly equal to μ . The mean of the distribution of Sample Means is equal to the mean of the popula-tion of scores, μ , and is called the expected value of M . ■ ■ The Standard Error of M So far, we have considered the shape and the central tendency of the distribution of Sample Means. To completely describe this distribution, we need one more characteristic, vari-ability.
- Robert M. Leekley(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
And serious inference generally requires more quantitative descriptors of what we have found in our samples. Hence, this chapter deals with such descriptors. You are already familiar with many of them. 3 .2 Measures of a Single Numeric Variable 3.2.1 Measures of Central Tendency Suppose you are given a set of numbers and asked to summarize them with just a single number. What would you do? Probably, you would calculate some sort of average. An average is a measure of the center. 44 Applied Statistics for Business and Economics It may be a sort of balance point in the data; it may be some measure of what is typical. 3.2.1.1 The Arithmetic Mean The arithmetic mean is what most people mean by an average. It is just the sum of the numbers divided by their count. It is the arithmetic mean, to distinguish it from other, special purpose means. But in practice, it is often shortened to just the mean, without any modifier. Figure 3.1 presents the formulas for calculating the mean of a variable, X . Of course, you already know how to calculate it. There is nothing new here except, perhaps, the symbols. The symbol for the mean of X in a population is μ X (“mu sub X ”). ∑ i is th symbol for a sum; X i represents an individual value of X , and N X repre-sents the number, or count, of the population. So the formula says simply to sum (∑) the individual values ( X i ) from the first ( i = 1) to the last ( N X ) and then divide by the number of individual values ( N X ). The symbol for the mean of X in a sample is X – (“ X bar”). Notice, the only difference in the formula is that N X , the population count, has been replaced with n X , the sample count. While to calculate μ X , you add up all the population values and divide by the population count, to calculate X – , you add up all the sample values and divide by the sample count. In other words, whether you have a population or a sample, you do exactly the same thing.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 3 Calculation and Basic Concepts of Means Arithmetic mean In mathematics and statistics, the arithmetic mean , often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space. The term arithmetic mean is preferred in mathematics and statistics because it helps distinguish it from other averages such as the geometric and harmonic mean. In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent. For example, per capita GDP gives an approximation of the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers . Notably, for skewed distributions, the arithmetic mean may not accord with one's notion of middle, and robust statistics such as the median may be a better description of central tendency. Definition Suppose we have sample space . Then the arithmetic mean A is defined via the equation . If the list is a statistical population, then the mean of that population is called a population mean . If the list is a statistical sample, we call the resulting statistic a Sample Mean . ________________________ WORLD TECHNOLOGIES ________________________ Motivating properties The arithmetic mean has several properties that make it useful, especially as a measure of central tendency. These include: • If numbers have mean X, then . Since x i − X is the distance from a given number to the mean, one way to interpret this property is as saying that the numbers to the left of the mean are balanced by the numbers to the right of the mean. The mean is the only single number for which the residuals defined this way sum to zero.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Calculation and Basic Concepts of Means Arithmetic mean In mathematics and statistics, the arithmetic mean , often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space. The term arithmetic mean is preferred in mathematics and statistics beca-use it helps distinguish it from other averages such as the geometric and harmonic mean. In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent. For example, per capita GDP gives an approximation of the arith-metic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers . Notably, for skewed distri-butions, the arithmetic mean may not accord with one's notion of middle, and robust statistics such as the median may be a better description of central tendency. Definition Suppose we have sample space . Then the arithmetic mean A is defined via the equation . If the list is a statistical population, then the mean of that population is called a popu-lation mean . If the list is a statistical sample, we call the resulting statistic a Sample Mean . ________________________ WORLD TECHNOLOGIES ________________________ Motivating properties The arithmetic mean has several properties that make it useful, especially as a measure of central tendency. These include: • If numbers have mean X, then . Since x i − X is the distance from a given number to the mean, one way to interpret this property is as saying that the numbers to the left of the mean are balanced by the numbers to the right of the mean. The mean is the only single number for which the residuals defined this way sum to zero.
- Alan R. Jones(Author)
- 2018(Publication Date)
- Routledge(Publisher)
As a consequence, we will use the term, Arithmetic Mean in order to differentiate it ( but not in the mathematical calculus sense, you will probably be relieved to hear) from the other Power Means. The tendency to refer to the Arithmetic Mean as simply ‘Average’ is reflected in Mic- rosoft Excel, where the in-built function for the Arithmetic Mean is AVERAGE( range). Definition 2-1 Arithmetic Mean or Average The Arithmetic Mean or Average of a set of numerical data values is a statistic calculated by summating the values of the individual terms and dividing by the number of terms in the set. Definition 2.1 Arithmetic Mean or Average The Arithmetic Mean or Average of a set of numerical data values is a statistic calculated by summating the values of the individual terms and dividing by the number of terms in the set. 20 | Measures of Central Tendency The Arithmetic Mean gives equal weighting to every single term in the set (or sample) of data. However, if we look at it another way, the Arithmetic Mean weights each unique value by the number of times it occurs. For this reason, the Arithmetic Mean is some- times called the ‘Expected Value’, i.e. the value we would expect based on the probability of any single value occurring. For the Formula-philes: Definition of the Arithmetic Mean Consider a range of n observations x 1 , x 2 , x 3 . . . x n The Arithmetic Mean of x, x : x x x x n n = + + + x x …+ 1 2 x x 3 Notationally, the Arithmetic Mean is: x n x i n i = = ∑ x 1 1 For the Formula-philes: Equivalence of the Arithmetic Mean and the Expected Value Consider a range of n observations x 1 , x 2 , x 3 . . . x n The Arithmetic Mean x of the data is: x n x i n i = = ∑ x 1 1 (1) Suppose that the observed data only take the val- ues X i , X 2 . . . X j where j ≤ n, and therefore one or more value of x i is repeated in the sequence x 1 , x 2 , x 3 . . . . x n . If values X 1 , X 2 . . . X j occur c i , c 2 . . . c j times respectively then: k j k c n k = ∑ c 1 .
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 2 Average An average , or central tendency of a data set is a measure of the middle value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items. These include arithmetic mean, the median and the mode. Other statistical measures such as the standard deviation and the range are called measures of spread and describe how spread out the data is. An average is a single value that is meant to typify a list of values. If all the numbers in the list are the same, then this number should be used. If the numbers are not the same, an easy way to get a representative value from a list is to randomly pick any number from the list. In the latter case, the average is calculated by combining the values from the set in a specific way and computing a single number as being the average of the set. The most common method is the arithmetic mean but there are many other types of central tendency, such as median (which is used most often when the distribution of the values is skewed with some small numbers of very high values, as seen with house prices or incomes). Calculation Arithmetic mean If n numbers are given, each number denoted by a i , where i = 1, ..., n , the arithmetic mean is the [sum] of the a i 's divided by n or The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. One may find that A = (2 + 8)/2 = 5. Switching the order of 2 and 8 to read 8 and 2 does not change the resulting value obtained for A. The mean 5 is not less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list for which we want an average,
- Robert Pagano(Author)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Central tendency and variability are the two characteristics of distributions that are most often quantified. In this chapter, we shall discuss the most important measures of these two characteristics. MEASURES OF CENTRAL TENDENCY The three most often used measures of central tendency are the arithmetic mean, the median, and the mode. The Arithmetic Mean You are probably already familiar with the arithmetic mean. It is the value you ordi-narily calculate when you average something. For example, if you wanted to know the average number of hours you studied per day for the past 5 days, you would add the hours you studied each day and divide by 5. In so doing, you would be calculating the arithmetic mean. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Measures of Central Tendency 81 definition ■ The arithmetic mean is defined as the sum of the scores divided by the number of scores. In equation form, X – X i N X 1 X 2 X 3 p X N N mean of sample or m X i N X 1 X 2 X 3 p X N N mean of population set of scores where X 1 … X N raw scores X – (read “X bar”) mean of a sample set of scores μ (read “mew”) mean of a population set of scores (read “sigma”) summation sign N number of scores Note that we use two symbols for the mean: X – if the scores are sample scores and μ (the Greek letter mu) if the scores are population scores. The computations, however, are the same regardless of whether the scores are sample or population scores. We shall use μ without any subscript to indicate that this is the mean of a population of raw scores.
eBook - ePubSocial Statistics
Managing Data, Conducting Analyses, Presenting Results
- Thomas J. Linneman(Author)
- 2017(Publication Date)
- Routledge(Publisher)
Samples with Sample Means a bit away from the population mean—2.2, 2.3, 2.7, 2.8—are still common; however, they are less common than the Sample Means of 2.4, 2.5, and 2.6. The Sample Means of 1.9 and 3.1 are even less likely to happen, but occasionally they did happen. And once, only once, did I draw a sample where the Sample Mean was 3.3. Oddly, it was the second sample I drew. The chances of drawing such a sample are very, very slim. Notice that we didn’t draw any samples for which the Sample Mean was as low as 1.2 or 1.3, or as high as 3.7 or 3.8. Think through why such occurrences did not happen. What would have had to fall into place for this to occur? To draw a sample with a mean of 1.2, I would have had to draw, say, a 1.0, 1.1, 1.2, 1.3, and 1.4. To draw a sample with a mean of 3.8, I would have had to draw, say, a 3.6, 3.7, 3.8, 3.9, and 4.0. Given the population of scores, the chances of such things happening are next to nothing. My hope is that all of this is giving you an intuitive feel for how Sample Means act: most of them will fall near the population mean, some will fall a bit away from the population mean, and a few will, by chance, fall quite far away from the population mean. This is simply how the laws of randomness work. What I have built above is a distribution of Sample Means, or, as it is more commonly called, a sampling distribution. Just as the chi-square distribution was the basis of the chi-square test, sampling distributions form the basis of much of the theory and practice on which mean-based inferential statistical techniques are built. With this sampling distribution, we can calculate the probability of pulling certain types of samples. To find such a probability, we simply take the total number of attempts (in this situation, 300, because we took 300 samples) and divide that into the number of times a particular event occurred
- Teresa Bradley(Author)
- 2014(Publication Date)
- Wiley(Publisher)
Parameters such as the population mean will be estimated by a Sample Mean, calculated from a random sample. At the outset, one might expect the Sample Mean to be close to the population mean but, in some cases, there will be a very large difference between them. In this chapter we will examine the variability in the values of the Sample Means (and proportions). We will show that under certain conditions, Sample Means are, in fact, Normally distributed and that the mean of all the Sample Means is the same as the population mean. Since the distribution of Sample Means in Normally distributed, we can calculate probabilities about the Sample Means and calculate intervals that contain a given percentage of all Sample Means. The confidence intervals and test hypothesis in later chapters are based on the probability distributions of sample statistics (such as Sample Means). 6.1.1 Sampling distributions of means and proportions For the remainder of the text we will study statistical inference. We will use sample statistics to make inference about population parameters. Therefore it is necessary to use different symbols to represent sample characteristics and the corresponding population characteristics. In general the population characteristics will be represented by letters from the Greek alphabet while sample characteristics will be represented by the corresponding letters from the English alphabet. The symbols for mean, variance and proportions are listed below in Table 6.1. Some reminders are also given on definitions and other essential background and terminology. Reminders A statistic is a sample characteristic, such as the Sample Mean x , the sample variance s 2 etc. A parameter is a population characteristic such as the population mean and variance; μ and σ 2 .
eBook - PDFQuantitative Techniques in Business, Management and Finance
A Case-Study Approach
- Umeshkumar Dubey, D P Kothari, G K Awari(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
47 Measures of Central Tendency µ = ∑ x/n = (22 + 23 + 21 + 24 + 4)/5 = 18.8 units When the mean units are calculated omitting the fifth worker (i.e. 5), the mean is 22.5 units. Thus, one extreme value (4) has affected the mean. Hence, it is more appropriate to calculate the mean excluding the extreme value in order to make it more representative. 3. In some cases, the arithmetic mean may give misleading impressions. For exam-ple, the average number of patients admitted in a hospital is 9.8 per day. Here, the mean is useful information, but it does not represent the actual item. 4. It is very difficult to find the actual mean (using µ = x/n ∑ ). 5. It can hardly be located by inspection. 6. It cannot be calculated even if one value is missing. 7. We cannot calculate the mean for a data set with open-ended classes at either end of the scale. A class that allows either the upper or lower end of a quantitative clas-sification scheme to be limitless is called as open-ended class. 3.4.2 Mathematical Properties of Arithmetic Mean Because the arithmetic is defined operationally, it has several useful mathematical proper-ties. Some of these are as follows: 1. The sum of deviations of the observations from the arithmetic mean is always zero. Symbolically, it is X X -( ) = ∑ 0 It is because of this property that the mean is characterised as a point of bal-ance; that is the sum of positive deviations from mean is equal to the sum of the negative deviations from mean. 2. The sum of squared deviations of the observations from the mean is minimum; that is the total of the squares of the deviations from any other value than the mean value will be greater than the total sum of squares of deviations from the mean. Symbolically, ∑ -( ) X X is a minimum 2 3. The arithmetic means of several sets of data may be combined into a single arith-metic mean for the combined sets of data.
eBook - PDFStatistics Using IBM SPSS
An Integrative Approach
- Sharon Lawner Weinberg, Sarah Knapp Abramowitz(Authors)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
Though we are studying the sample, our real interest in inferential statistics is the population, and the conclusions we will draw are always about it. We can describe measures taken on populations of things in the same ways that we can describe measures taken on samples of things. For example, if we could get data from the entire population, we could compute all measures of location, variability, and relationship (e.g., the mean, standard deviation, and correlation) just as we can on a sample taken from that population. When computed for populations, the values of such descriptive measures are called parameters ; when computed for samples, they are called statistics . In inferential statistics, we use statistics computed on a sample to draw conclusions about unknown parameters in the population. In order to make it clear whether we are talking about a measure taken on a sample or on a population, we denote statistics by Roman letters and parameters by Greek letters. Table 9.1 contains examples of these symbols. As another example, consider that we want to find the mean (average) number of hours eighth graders in the United States spend watching television per week. In order to find exactly what we want, the population mean, µ , we would somehow have to obtain the number of hours of television viewing for each member of the population. This is obviously impractical. Instead we might pick a sample of, say, 1,000 eighth graders in the New York THE ROLE OF SAMPLING IN INFERENTIAL STATISTICS 246 metropolitan area, find the mean number of hours of television watched in the sample, and then use our calculated value of the Sample Mean, X , as an estimate of the unknown population mean µ . Do you think the procedure we have just outlined is a good one for estimating the mean number of hours of television viewing, µ , of the population described? Actually, there are several things wrong with it.
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