Chapter 1. Descriptive Statistics and Frequency Distributions

This chapter is about describing populations and samples, a subject known as descriptive statistics. This will all make more sense if you keep in mind that the information you want to produce is a description of the population or sample as a whole, not a description of one member of the population. The first topic in this chapter is a discussion of distributions, essentially pictures of populations (or samples). Second will be the discussion of descriptive statistics. The topics are arranged in this order because the descriptive statistics can be thought of as ways to describe the picture of a population, the distribution.

Distributions

The first step in turning data into information is to create a distribution. The most primitive way to present a distribution is to simply list, in one column, each value that occurs in the population and, in the next column, the number of times it occurs. It is customary to list the values from lowest to highest. This simple listing is called a frequency distribution. A more elegant way to turn data into information is to draw a graph of the distribution. Customarily, the values that occur are put along the horizontal axis and the frequency of the value is on the vertical axis.

Ann is the equipment manager for the Chargers athletic teams at Camosun College, located in Victoria, British Columbia. She called the basketball and volleyball team managers and collected the following data on sock sizes used by their players. Ann found out that last year the basketball team used 14 pairs of size 7 socks, 18 pairs of size 8, 15 pairs of size 9, and 6 pairs of size 10 were used. The volleyball team used 3 pairs of size 6, 10 pairs of size 7, 15 pairs of size 8, 5 pairs of size 9, and 11 pairs of size 10. Ann arranged her data into a distribution and then drew a graph called a histogram. Ann could have created a relative frequency distribution as well as a frequency distribution. The difference is that instead of listing how many times each value occurred, Ann would list what proportion of her sample was made up of socks of each size.

You can use the Excel template below (Figure 1.1) to see all the histograms and frequencies she has created. You may also change her numbers in the yellow cells to see how the graphs will change automatically.

Figure 1.1 Interactive Excel Template of a Histogram – see Appendix 1.

Notice that Ann has drawn the graphs differently. In the first graph, she has used bars for each value, while on the second, she has drawn a point for the relative frequency of each size, and then “connected the dots”. While both methods are correct, when you have values that are continuous, you will want to do something more like the “connect the dots” graph. Sock sizes are discrete, they only take on a limited number of values. Other things have continuous values; they can take on an infinite number of values, though we are often in the habit of rounding them off. An example is how much students weigh. While we usually give our weight in whole kilograms in Canada (“I weigh 60 kilograms”), few have a weight that is exactly so many kilograms. When you say “I weigh 60”, you actually mean that you weigh between 59 1/2 and 60 1/2 kilograms. We are heading toward a graph of a distribution of a continuous variable where the relative frequency of any exact value is very small, but the relative frequency of observations between two values is measurable. What we want to do is to get used to the idea that the total area under a “connect the dots” relative frequency graph, from the lowest to the highest possible value, is one. Then the part of the area under the graph between two values is the relative frequency of observations with values within that range. The height of the line above any particular value has lost any direct meaning, because it is now the area under the line between two values that is the relative frequency of an observation between those two values occurring.

You can get some idea of how this works if you go back to the bar graph of the distribution of sock sizes, but draw it with relative frequency on the vertical axis. If you arbitrarily decide that each bar has a width of one, then the area under the curve between 7.5 and 8.5 is simply the height times the width of the bar for sock size 8: .3510*1. If you wanted to find the relative frequency of sock sizes between 6.5 and 8.5, you could simply add together the area of the bar for size 7 (that’s between 6.5 and 7.5) and the bar for size 8 (between 7.5 and 8.5).

Descriptive statistics

Now that you see how a distribution is created, you are ready to learn how to describe one. There are two main things that need to be described about a distribution: its location and its shape. Generally, it is best to give a single measure as the description of the location and a single measure as the description of the shape.

Mean

To describe the location of a distribution, statisticians use a typical value from the distribution. There are a number of different ways to find the typical value, but by far the most used is the arithmetic mean, usually simply called the mean. You already know how to find the arithmetic mean, you are just used to calling it the average. Statisticians use average more generally — the arithmetic mean is one of a number of different averages. Look at the formula for the arithmetic mean:

[latex]\mu = \dfrac{\sum{x}}{N}[/latex]

All you do is add up all of the members of the population, [latex]\sum{x}[/latex], and divide by how many members there are, N. The only trick is to remember that if there is more than one member of the population with a certain value, to add that value once for every member that has it. To reflect this, the equation for the mean sometimes is written:

[latex]\mu = \dfrac{\sum{f_i(x_i)}}{N}[/latex]

where f_i is the frequency of members of the population with the value x_i.

This is really the same formula as above. If there are seven members with a value of ten, the first formula would have you add seven ten times. The second formula simply has you multiply seven by ten — the same thing as adding together ten sevens.

Other measures of location are the median and the mode. The median is the value of the member of the population that is in the middle when the members are sorted from smallest to largest. Half of the members of the population have values higher than the median, and half have values lower. The median is a better measure of location if there are one or two members of the population that are a lot larger (or a lot smaller) than all the rest. Such extreme values can make the mean a poor measure of location, while they have little effect on the median. If there are an odd number of members of the population, there is no problem finding which member has the median value. If there are an even number of members of the population, then there is no single member in the middle. In that case, just average together the values of the two members that share the middle.

The third common measure of location is the mode. If you have arranged the population into a frequency or relative frequency distribution, the mode is easy to find because it is the value that occurs most often. While in some sense, the mode is really the most typical member of the population, it is often not very near the middle of the population. You can also have multiple modes. I am sure you have heard someone say that “it was a bimodal distribution“. That simply means that there were two modes, two values that occurred equally most often.

If you think about it, you should not be surprised to learn that for bell-shaped distributions, the mean, median, and mode will be equal. Most of what statisticians do when describing or inferring the location of a population is done with the mean. Another thing to think about is using a spreadsheet program, like Microsoft Excel, when arranging data into a frequency distribution or when finding the median or mode. By using the sort and distribution commands in 1-2-3, or similar commands in Excel, data can quickly be arranged in order or placed into value classes and the number in each class found. Excel also has a function, =AVERAGE(…), for finding the arithmetic mean. You can also have the spreadsheet program draw your frequency or relative frequency distribution.

One of the reasons that the arithmetic mean is the most used measure of location is because the mean of a sample is an unbiased estimator of the population mean. Because the sample mean is an unbiased estimator of the population mean, the sample mean is a good way to make an inference about the population mean. If you have a sample from a population, and you want to guess what the mean of that population is, you can legitimately guess that the population mean is equal to the mean of your sample. This is a legitimate way to make this inference because the mean of all the sample means equals the mean of the population, so if you used this method many times to infer the population mean, on average you’d be correct.

All of these measures of location can be found for samples as well as populations, using the same formulas. Generally, μ is used for a population mean, and x is used for sample means. Upper-case N, really a Greek nu, is used for the size of a population, while lower case n is used for sample size. Though it is not universal, statisticians tend to use the Greek alphabet for population characteristics and the Roman alphabet for sample characteristics.

Measuring population shape

Measuring the shape of a distribution is more difficult. Location has only one dimension (“where?”), but shape has a lot of dimensions. We will talk about two,and you will find that most of the time, only one dimension of shape is measured. The two dimensions of shape discussed here are the width and symmetry of the distribution. The simplest way to measure the width is to do just that—the range is the distance between the lowest and highest members of the population. The range is obviously affected by one or two population members that are much higher or lower than all the rest.

The most common measures of distribution width are the standard deviation and the variance. The standard deviation is simply the square root of the variance, so if you know one (and have a calculator that does squares and square roots) you know the other. The standard deviation is just a strange measure of the mean distance between the members of a population and the mean of the population. This is easiest to see if you start out by looking at the formula for the variance:

[latex]\sigma^2 = \dfrac{\sum{(x-\mu)^2}}{N}[/latex]

Look at the numerator. To find the variance, the first step (after you have the mean, μ) is to take each member of the population, and find the difference between its value and the mean; you should have N differences. Square each of those, and add them together, dividing the sum by N, the number of members of the population. Since you find the mean of a group of things by adding them together and then dividing by the number in the group, the variance is simply the mean of the squared distances between members of the population and the population mean.

Notice that this is the formula for a population characteristic, so we use the Greek σ and that we write the variance as σ², or sigma square because the standard deviation is simply the square root of the variance, its symbol is simply sigma, σ.

One of the things statisticians have discovered is that 75 per cent of the members of any population are within two standard deviations of the mean of the population. This is known as Chebyshev’s theorem. If the mean of a population of shoe sizes is 9.6 and the standard deviation is 1.1, then 75 per cent of the shoe sizes are between 7.4 (two standard deviations below the mean) and 11.8 (two standard deviations above the mean). This same theorem can be stated in probability terms: the probability that anything is within two standard deviations of the mean of its population is .75.

It is important to be careful when dealing with variances and standard deviations. In later chapters, there are formulas using the variance, and formulas using the standard deviation. Be sure you know which one you are supposed to be using. Here again, spreadsheet programs will figure out the standard deviation for you. In Excel, there is a function, =STDEVP(…), that does all of the arithmetic. Most calculators will also compute the standard deviation. Read the little instruction booklet, and find out how to have your calculator do the numbers before you do any homework or have a test.

The other measure of shape we will discuss here is the measure of skewness. Skewness is simply a measure of whether or not the distribution is symmetric or if it has a long tail on one side, but not the other. There are a number of ways to measure skewness, with many of the measures based on a formula much like the variance. The formula looks a lot like that for the variance, except the distances between the members and the population mean are cubed, rather than squared, before they are added together:

[latex]sk = \dfrac{\sum{(x-\mu)^3}}{N}[/latex]

At first, it might not seem that cubing rather than squaring those distances would make much difference. Remember, however, that when you square either a positive or negative number, you get a positive number, but when you cube a positive, you get a positive and when you cube a negative you get a negative. Also remember that when you square a number, it gets larger, but that when you cube a number, it gets a whole lot larger. Think about a distribution with a long tail out to the left. There are a few members of that population much smaller than the mean, members for which (x – μ) is large and negative. When these are cubed, you end up with some really big negative numbers. Because there are no members with such large, positive (x – μ), there are no corresponding really big positive numbers to add in when you sum up the (x – μ)³, and the sum will be negative. A negative measure of skewness means that there is a tail out to the left, a positive measure means a tail to the right. Take a minute and convince yourself that if the distribution is symmetric, with equal tails on the left and right, the measure of skew is zero.

To be really complete, there is one more thing to measure, kurtosis or peakedness. As you might expect by now, it is measured by taking the distances between the members and the mean and raising them to the fourth power before averaging them together.

Measuring sample shape

Measuring the location of a sample is done in exactly the way that the location of a population is done. However, measuring the shape of a sample is done a little differently than measuring the shape of a population. The reason behind the difference is the desire to have the sample measurement serve as an unbiased estimator of the population measurement. If we took all of the possible samples of a certain size, n, from a population and found the variance of each one, and then found the mean of those sample variances, that mean would be a little smaller than the variance of the population.

You can see why this is so if you think it through. If you knew the population mean, you could find [latex]\sum{\dfrac{(x-\mu)^2}{n}}[/latex] for each sample, and have an unbiased estimate for σ². However, you do not know the population mean, so you will have to infer it. The best way to infer the population mean is to use the sample mean x. The variance of a sample will then be found by averaging together all of the [latex]\sum{\dfrac{(x-\bar{x})^2}{n}}[/latex].

The mean of a sample is obviously determined by where the members of that sample lie. If you have a sample that is mostly from the high (or right) side of a population’s distribution, then the sample mean will almost for sure be greater than the population mean. For such a sample, [latex]\sum{\dfrac{(x-\bar{x})^2}{n}}[/latex] would underestimate σ². The same is true for samples that are mostly from the low (or left) side of the population. If you think about what kind of samples will have [latex]\sum{\dfrac{(x-\bar{x})^2}{n}}[/latex] that is greater than the population σ², you will come to the realization that it is only those samples with a few very high members and a few very low members — and there are not very many samples like that. By now you should have convinced yourself that [latex]\sum{\dfrac{(x-\bar{x})^2}{n}}[/latex] will result in a biased estimate of σ². You can see that, on average, it is too small.

How can an unbiased estimate of the population variance, σ², be found? If [latex]\sum{\dfrac{(x-\bar{x})^2}{n}}[/latex] is on average too small, we need to do something to make it a little bigger. We want to keep the [latex]\sum{(x-\bar{x})^2}[/latex], but if we divide it by something a little smaller, the result will be a little larger. Statisticians have found out that the following way to compute the sample variance results in an unbiased estimator of the population variance:

[latex]s^2 = \dfrac{\sum{(x-\bar{x})^2}}{n-1}[/latex]

If we took all of the possible samples of some size, n, from a population, and found the sample variance for each of those samples, using this formula, the mean of those sample variances would equal the population variance, σ².

Note that we use s² instead of σ², and n instead of N (really nu, not en) since this is for a sample and we want to use the Roman letters rather than the Greek letters, which are used for populations.

There is another way to see why you divide by n-1. We also have to address something called degrees of freedom before too long, and the degrees of freedom are the key in the other explanation. As we go through this explanation, you should be able to see that the two explanations are related.

Imagine that you have a sample with 10 members, n=10, and you want to use it to estimate the variance of the population from which it was drawn. You write each of the 10 values on a separate scrap of paper. If you know the population mean, you could start by computing all 10 (x – μ)². However, in the usual case, you do not know μ, and you must start by finding x from the values on the 10 scraps to use as an estimate of m. Once you have found x, you could lose any one of the 10 scraps and still be able to find the value that was on the lost scrap from the other 9 scraps. If you are going to use x in the formula for sample variance, only 9 (or n-1) of the x’s are free to take on any value. Because only n-1 of the x’s can vary freely, you should divide [latex]\sum{(x-\bar{x})^2}[/latex] by n-1, the number of (x’s) that are really free. Once you use x in the formula for sample variance, you use up one degree of freedom, leaving only n-1. Generally, whenever you use something you have previously computed from a sample within a formula, you use up a degree of freedom.

A little thought will link the two explanations. The first explanation is based on the idea that x, the estimator of μ, varies with the sample. It is because x varies with the sample that a degree of freedom is used up in the second explanation.

The sample standard deviation is found simply by taking the square root of the sample variance:

[latex]s=\surd[\dfrac{\sum{(x-\bar{x}})^2}{n-1}][/latex]

While the sample variance is an unbiased estimator of population variance, the sample standard deviation is not an unbiased estimator of the population standard deviation — the square root of the average is not the same as the average of the square roots. This causes statisticians to use variance where it seems as though they are trying to get at standard deviation. In general, statisticians tend to use variance more than standard deviation. Be careful with formulas using sample variance and standard deviation in the following chapters. Make sure you are using the right one. Also note that many calculators will find standard deviation using both the population and sample formulas. Some use σ and s to show the difference between population and sample formulas, some use s_n and s_n-1 to show the difference.

If Ann wanted to infer what the population distribution of volleyball players’ sock sizes looked like she could do so from her sample. If she is going to send volleyball coaches packages of socks for the players to try, she will want to have the packages contain an assortment of sizes that will allow each player to have a pair that fits. Ann wants to infer what the distribution of volleyball players’ sock sizes looks like. She wants to know the mean and variance of that distribution. Her data, again, are shown in Table 1.1.

The mean sock size can be found:
[latex]=\dfrac{3*6+24*7+33*8+20*9+17*10}{97} = 8.25[/latex]

To find the sample standard deviation, Ann decides to use Excel. She lists the sock sizes that were in the sample in column A (see Table 1.2) , and the frequency of each of those sizes in column B. For column C, she has the computer find for each of [latex]\sum{(x-\bar{x})^2}[/latex] the sock sizes, using the formula (A1-8.25)² in the first row, and then copying it down to the other four rows. In D1, she multiplies C1, by the frequency using the formula =B1*C1, and copying it down into the other rows. Finally, she finds the sample standard deviation by adding up the five numbers in column D and dividing by n-1 = 96 using the Excel formula =sum(D1:D5)/96. The spreadsheet appears like this when she is done:

Ann now has an estimate of the variance of the sizes of socks worn by basketball and volleyball players, 1.22. She has inferred that the population of Chargers players’ sock sizes has a mean of 8.25 and a variance of 1.22.

Ann’s collected data can simply be added to the following Excel template. The calculations of both variance and standard deviation have been shown below. You can change her numbers to see how these two measures change.

Figure 1.2 Interactive Excel Template to Calculate Variance and Standard Deviation – see Appendix 1.

Summary

To describe a population you need to describe the picture or graph of its distribution. The two things that need to be described about the distribution are its location and its shape. Location is measured by an average, most often the arithmetic mean. The most important measure of shape is a measure of dispersion, roughly width, most often the variance or its square root the standard deviation.

Samples need to be described, too. If all we wanted to do with sample descriptions was describe the sample, we could use exactly the same measures for sample location and dispersion that are used for populations. However, we want to use the sample describers for dual purposes: (a) to describe the sample, and (b) to make inferences about the description of the population that sample came from. Because we want to use them to make inferences, we want our sample descriptions to be unbiased estimators. Our desire to measure sample dispersion with an unbiased estimator of population dispersion means that the formula we use for computing sample variance is a little different from the one used for computing population variance.

Chapter 2. The Normal and t-Distributions

The normal distribution is simply a distribution with a certain shape. It is normal because many things have this same shape. The normal distribution is the bell-shaped distribution that describes how so many natural, machine-made, or human performance outcomes are distributed. If you ever took a class when you were “graded on a bell curve”, the instructor was fitting the class’s grades into a normal distribution—not a bad practice if the class is large and the tests are objective, since human performance in such situations is normally distributed. This chapter will discuss the normal distribution and then move on to a common sampling distribution, the t-distribution. The t-distribution can be formed by taking many samples (strictly, all possible samples) of the same size from a normal population. For each sample, the same statistic, called the t-statistic, which we will learn more about later, is calculated. The relative frequency distribution of these t-statistics is the t-distribution. It turns out that t-statistics can be computed a number of different ways on samples drawn in a number of different situations and still have the same relative frequency distribution. This makes the t-distribution useful for making many different inferences, so it is one of the most important links between samples and populations used by statisticians. In between discussing the normal and t-distributions, we will discuss the central limit theorem. The t-distribution and the central limit theorem give us knowledge about the relationship between sample means and population means that allows us to make inferences about the population mean.

The way the t-distribution is used to make inferences about populations from samples is the model for many of the inferences that statisticians make. Since you will be learning to make inferences like a statistician, try to understand the general model of inference making as well as the specific cases presented. Briefly, the general model of inference-making is to use statisticians’ knowledge of a sampling distribution like the t-distribution as a guide to the probable limits of where the sample lies relative to the population. Remember that the sample you are using to make an inference about the population is only one of many possible samples from the population. The samples will vary, some being highly representative of the population, most being fairly representative, and a few not being very representative at all. By assuming that the sample is at least fairly representative of the population, the sampling distribution can be used as a link between the sample and the population so you can make an inference about some characteristic of the population.

These ideas will be developed more later on. The immediate goal of this chapter is to introduce you to the normal distribution, the central limit theorem, and the t-distribution.

Normal Distributions

Normal distributions are bell-shaped and symmetric. The mean, median, and mode are equal. Most of the members of a normally distributed population have values close to the mean—in a normal population 96 per cent of the members (much better than Chebyshev’s 75 per cent) are within 2 σ of the mean.

Statisticians have found that many things are normally distributed. In nature, the weights, lengths, and thicknesses of all sorts of plants and animals are normally distributed. In manufacturing, the diameter, weight, strength, and many other characteristics of human- or machine-made items are normally distributed. In human performance, scores on objective tests, the outcomes of many athletic exercises, and college student grade point averages are normally distributed. The normal distribution really is a normal occurrence.

If you are a skeptic, you are wondering how can GPAs and the exact diameter of holes drilled by some machine have the same distribution—they are not even measured with the same units. In order to see that so many things have the same normal shape, all must be measured in the same units (or have the units eliminated)—they must all be standardized. Statisticians standardize many measures by using the standard deviation. All normal distributions have the same shape because they all have the same relative frequency distribution when the values for their members are measured in standard deviations above or below the mean.

Using the customary Canadian system of measurement, if the weight of pet dogs is normally distributed with a mean of 10.8 kilograms and a standard deviation of 2.3 kilograms and the daily sales at The First Brew Expresso Cafe are normally distributed with μ = $341.46 and σ = $53.21, then the same proportion of pet dogs weigh between 8.5 kilograms (μ – 1σ) and 10.8 kilograms (μ) as the proportion of daily First Brew sales that lie between μ – 1σ ($288.25) and μ ($341.46). Any normally distributed population will have the same proportion of its members between the mean and one standard deviation below the mean. Converting the values of the members of a normal population so that each is now expressed in terms of standard deviations from the mean makes the populations all the same. This process is known as standardization, and it makes all normal populations have the same location and shape.

This standardization process is accomplished by computing a z-score for every member of the normal population. The z-score is found by:

[latex]z = (x - \mu)/\sigma[/latex]

This converts the original value, in its original units, into a standardized value in units of standard deviations from the mean. Look at the formula. The numerator is simply the difference between the value of this member of the population x, and the mean of the population μ. It can be measured in centimeters, or points, or whatever. The denominator is the standard deviation of the population, σ, and it is also measured in centimetres, or points, or whatever. If the numerator is 15 cm and the standard deviation is 10 cm, then the z will be 1.5. This particular member of the population, one with a diameter 15 cm greater than the mean diameter of the population, has a z-value of 1.5 because its value is 1.5 standard deviations greater than the mean. Because the mean of the x’s is μ, the mean of the z-scores is zero.

We could convert the value of every member of any normal population into a z-score. If we did that for any normal population and arranged those z-scores into a relative frequency distribution, they would all be the same. Each and every one of those standardized normal distributions would have a mean of zero and the same shape. There are many tables that show what proportion of any normal population will have a z-score less than a certain value. Because the standard normal distribution is symmetric with a mean of zero, the same proportion of the population that is less than some positive z is also greater than the same negative z. Some values from a standard normal table appear in Table 2.1

You can also use the interactive cumulative standard normal distributions illustrated in the Excel template in Figure 2.1. The graph on the top calculates the z-value if any probability value is entered in the yellow cell. The graph on the bottom computes the probability of z for any given z-value in the yellow cell. In either case, the plot of the appropriate standard normal distribution will be shown with the cumulative probabilities in yellow or purple.

Figure 2.1 Interactive Excel Template for Cumulative Standard Normal Distributions – see Appendix 2.

The production manager of a beer company located in Delta, BC, has asked one of his technicians, Kevin, “How much does a pack of 24 beer bottles usually weigh?” Kevin asks the people in quality control what they know about the weight of these packs and is told that the mean weight is 16.32 kilograms with a standard deviation of .87 kilograms. Kevin decides that the production manager probably wants more than the mean weight and decides to give his boss the range of weights within which 95% of packs of 24 beer bottles falls. Kevin sees that leaving 2.5% (.025 ) in the left tail and 2.5% (.025) in the right tail will leave 95% (.95) in the middle. He assumes that the pack weights are normally distributed, a reasonable assumption for a machine-made product, and consulting a standard normal table, he sees that .975 of the members of any normal population have a z-score less than 1.96 and that .975 have a z-score greater than -1.96, so .95 have a z-score between ±1.96.

Now that he knows that 95% of the 24 packs of beer bottles will have a weight with a z-score between ±1.96, Kevin can translate those z’s. By solving the equation for both +1.96 and -1.96, he will find the boundaries of the interval within which 95% of the weights of the packs fall:

[latex]1.96=(x-16.32)/.87[/latex]

Solving for x, Kevin finds that the upper limit is 18.03 kilograms. He then solves for z=-1.96:

[latex]-1.96=(x-16.32)/.87[/latex]

He finds that the lower limit is 14.61 kilograms. He can now go to his manager and tell him: “95% of the packs of 24 beer bottles weigh between 14.61 and 18.03 kilograms.”

The central limit theorem

If this was a statistics course for math majors, you would probably have to prove this theorem. Because this text is designed for business and other non-math students, you will only have to learn to understand what the theorem says and why it is important. To understand what it says, it helps to understand why it works. Here is an explanation of why it works.

The theorem is about sampling distributions and the relationship between the location and shape of a population and the location and shape of a sampling distribution generated from that population. Specifically, the central limit theorem explains the relationship between a population and the distribution of sample means found by taking all of the possible samples of a certain size from the original population, finding the mean of each sample, and arranging them into a distribution.

The sampling distribution of means is an easy concept. Assume that you have a population of x’s. You take a sample of n of those x’s and find the mean of that sample, giving you one x. Then take another sample of the same size, n, and find its x. Do this over and over until you have chosen all possible samples of size n. You will have generated a new population, a population of x’s. Arrange this population into a distribution, and you have the sampling distribution of means. You could find the sampling distribution of medians, or variances, or some other sample statistic by collecting all of the possible samples of some size, n, finding the median, variance, or other statistic about each sample, and arranging them into a distribution.

The central limit theorem is about the sampling distribution of means. It links the sampling distribution of x’s with the original distribution of x’s. It tells us that:

(1) The mean of the sample means equals the mean of the original population, μ_x = μ. This is what makes x an unbiased estimator of μ.

(2) The distribution of x’s will be bell-shaped, no matter what the shape of the original distribution of x’s.

This makes sense when you stop and think about it. It means that only a small portion of the samples have means that are far from the population mean. For a sample to have a mean that is far from μ_x, almost all of its members have to be from the right tail of the distribution of x’s, or almost all have to be from the left tail. There are many more samples with most of their members from the middle of the distribution, or with some members from the right tail and some from the left tail, and all of those samples will have an x close to μ_x.

(3a) The larger the samples, the closer the sampling distribution will be to normal, and

(3b) if the distribution of x’s is normal, so is the distribution of x’s.

These come from the same basic reasoning as (2), but would require a formal proof since normal distribution is a mathematical concept. It is not too hard to see that larger samples will generate a “more bell-shaped” distribution of sample means than smaller samples, and that is what makes (3a) work.

(4) The variance of the x’s is equal to the variance of the x’s divided by the sample size, or:

[latex]\sigma^2_x = \sigma^2/n[/latex]

therefore the standard deviation of the sampling distribution is:

[latex]\sigma_x = \sigma/\sqrt{n}[/latex]

While it is a difficult to see why this exact formula holds without going through a formal proof, the basic idea that larger samples yield sampling distributions with smaller standard deviations can be understood intuitively. If [latex]\sigma_{\bar{x}} = \sigma_{\bar{x}}/\sqrt{n}[/latex]  then [latex]\sigma_{\bar{x}} < \sigma_A[/latex]. Furthermore, when the sample size n rises, σ²_x gets smaller. This is because it becomes more unusual to get a sample with an x that is far from μ as n gets larger. The standard deviation of the sampling distribution includes an (x - μ) for each, but remember that there are not many x’s that are as far from μ as there are x’s that are far from μ, and as n grows there are fewer and fewer samples with an x far from μ. This means that there are not many (x - μ) that are as large as quite a few (x - μ) are. By the time you square everything, the average is going to be much smaller that the average (x – μ)², so [latex]\sigma_{\bar{x}}[/latex] is going to be smaller than σ_x. If the mean volume of soft drink in a population of 355 mL cans is 360 mL with a variance of 5 (and a standard deviation of  2.236), then the sampling distribution of means of samples of nine cans will have a mean of 360 mL and a variance of 5/9=.556 (and a standard deviation of 2.236/3=.745).

You can also use the interactive Excel template in Figure 2.2 that illustrates the central limit theorem. Simply double click on the yellow cell in the sheet called CLT(n=5) or in the yellow cell of the sheet called CLT(n=15), and then click enter. Do not try to change the formula in these yellow cells. This will automatically take a sample from the population distribution and recreate the associated sampling distribution of x. You can repeat this process by double clicking on the yellow cell to see that regardless of the population distribution, the sampling distribution of x is approximately normal. You will also realize that the mean of the population, and the sampling distribution of x are always the same.

Figure 2.2 Interactive Excel Template for Illustrating the Central Limit Theorem - see Appendix 2.

Following this same line of reasoning, you can see in the Figure 2.2 template that when you do the resampling processes with n=5 and then n=15, the sampling error becomes smaller. You can also observe, when you change the sample size from 5 to 15 (moving from sheet CLT(n=15) to CLT(n=5)), that as the sample size gets larger, the variance and standard deviation of the sampling distribution get smaller. Just remember that as sample size grows, samples with an x that is far from μ get rarer and rarer, so that the average (x - μ)² gets smaller. The average (x - μ)² is the variance.

Back to the soft drink example. If larger samples of soft drink bottles are taken, say samples of 16, even fewer of the samples will have means that are very far from the mean of 360 mL. The variance of the sampling distribution when n=16 will therefore be smaller. According to what you have just learned, the variance will be only 5/16=.3125 (and the standard deviation will be 2.236/4=.559). The formula matches what logically is happening; as the samples get bigger, the probability of getting a sample with a mean that is far away from the population mean gets smaller, so the sampling distribution of means gets narrower and the variance (and standard deviation) get smaller. In the formula, you divide the population variance by the sample size to get the sampling distribution variance. Since bigger samples means dividing by a bigger number, the variance falls as sample size rises. If you are using the sample mean to infer the population mean, using a bigger sample will increase the probability that your inference is very close to correct because more of the sample means are very close to the population mean. There is obviously a trade-off here. The reason you wanted to use statistics in the first place was to avoid having to go to the bother and expense of collecting lots of data, but if you collect more data, your statistics will probably be more accurate.

The t-distribution

The central limit theorem tells us about the relationship between the sampling distribution of means and the original population. Notice that if we want to know the variance of the sampling distribution we need to know the variance of the original population. You do not need to know the variance of the sampling distribution to make a point estimate of the mean, but other, more elaborate, estimation techniques require that you either know or estimate the variance of the population. If you reflect for a moment, you will realize that it would be strange to know the variance of the population when you do not know the mean. Since you need to know the population mean to calculate the population variance and standard deviation, the only time when you would know the population variance without the population mean are examples and problems in textbooks. The usual case occurs when you have to estimate both the population variance and mean. Statisticians have figured out how to handle these cases by using the sample variance as an estimate of the population variance (and using that to estimate the variance of the sampling distribution). Remember that is an unbiased estimator of σ². Remember, too, that the variance of the sampling distribution of means is related to the variance of the original population according to the equation:

[latex]\sigma^2_x = \sigma^2/n[/latex]

So the estimated standard deviation of a sampling distribution of means is:

[latex]% estimated\;\sigma_x = s/\sqrt{n}[/latex]

Following this thought, statisticians found that if they took samples of a constant size from a normal population, computed a statistic called a t-score for each sample, and put those into a relative frequency distribution, the distribution would be the same for samples of the same size drawn from any normal population. The shape of this sampling distribution of t’s varies somewhat as sample size varies, but for any n, it’s always the same. For example, for samples of 5, 90% of the samples have t-scores between -1.943 and +1.943, while for samples of 15, 90% have t-scores between ± 1.761. The bigger the samples, the narrower the range of scores that covers any particular proportion of the samples. That t-score is computed by the formula:

[latex]t=(x-\mu)/(s/\sqrt{n})[/latex]

By comparing the formula for the t-score with the formula for the z-score, you will be able to see that the t is just an estimated z. Since there is one t-score for each sample, the t is just another sampling distribution. It turns out that there are other things that can be computed from a sample that have the same distribution as this t. Notice that we’ve used the sample standard deviation, s, in computing each t-score. Since we’ve used s, we’ve used up one degree of freedom. Because there are other useful sampling distributions that have this same shape, but use up various numbers of degrees of freedom, it is the usual practice to refer to the t-distribution not as the distribution for a particular sample size, but as the distribution for a particular number of degrees of freedom (df). There are published tables showing the shapes of the t-distributions, and they are arranged by degrees of freedom so that they can be used in all situations.

Looking at the formula, you can see that the mean t-score will be zero since the mean x equals μ. Each t-distribution is symmetric, with half of the t-scores being positive and half negative because we know from the central limit theorem that the sampling distribution of means is normal, and therefore symmetric, when the original population is normal.

An excerpt from a typical t-table is shown in Table 2.2. Note that there is one line each for various degrees of freedom. Across the top are the proportions of the distributions that will be left out in the tail--the amount shaded in the picture. The body of the table shows which t-score divides the bulk of the distribution of t’s for that df from the area shaded in the tail, which t-score leaves that proportion of t’s to its right. For example, if you chose all of the possible samples with 9 df, and found the t-score for each, .025 (2 1/2 %) of those samples would have t-scores greater than 2.262, and .975 would have t-scores less than 2.262.

In Table 2.2, a sampling of a student’s t-table, it shows the probability of exceeding the value in the body.  With 5 df, there is a .05 probability that a sample will have a t-score > 2.015.

For a more interactive t-table, along with the t-distribution, follow the Excel template in Figure 2.3. You can simply change the values in the yellow cells to see the cut-off point of the t-table, and its associated distribution.

Figure 2.3 Interactive Excel Template of a t-Table - see Appendix 2.

Since the t-distributions are symmetric, if 2 1/2% (.025) of the t’s with 9 df are greater than 2.262, then 2 1/2% are less than -2.262. The middle 95% (.95) of the t’s, when there are 9 df, are between -2.262 and +2.262. The middle .90 of t-scores when there are 14 df are between ±1.761, because -1.761 leaves .05 in the left tail and +1.761 leaves .05 in the right tail. The t-distribution gets closer and closer to the normal distribution as the number of degrees of freedom rises. As a result, the last line in the t-table, for infinity df, can also be used to find the z-scores that leave different proportions of the sample in the tail.

What could Kevin have done if he had been asked, "How much does a pack of 24 beer bottles weigh?" and could not easily find good data on the population? Since he knows statistics, he could take a sample and make an inference about the population mean. Because the distribution of weights of packs of 24 beer bottles is the result of a manufacturing process, it is almost certainly normal. The characteristics of almost every manufactured product are normally distributed. In a manufacturing process, even one that is precise and well controlled, each individual piece varies slightly as the temperature varies somewhat, the strength of the power varies as other machines are turned on and off, the consistency of the raw material varies slightly, and dozens of other forces that affect the final outcome vary slightly. Most of the packs, or bolts, or whatever is being manufactured, will be very close to the mean weight, or size, with just as many a little heavier or larger as there are a little lighter or smaller. Even though the process is supposed to be producing a population of "identical" items, there will be some variation among them. This is what causes so many populations to be normally distributed. Because the distribution of weights is normal, Kevin can use the t-table to find the shape of the distribution of sample t-scores. Because he can use the t-table to tell him about the shape of the distribution of sample t-scores, he can make a good inference about the mean weight of a pack of 24 beer bottles. This is how he could make that inference:

STEP 1. Take a sample of n, say 15, packs of beer bottles and carefully weigh each pack.

STEP 2. Find x and s for the sample.

STEP 3 (where the tricky part starts). Look at the t-table, and find the t-scores that leave some proportion, say .95, of sample t’s with n-1 df in the middle.

STEP 4 (the heart of the tricky part). Assume that the sample has a t-score that is in the middle part of the distribution of t-scores.

STEP 5 (the arithmetic). Take the x, s, n, and t’s from the t-table, and set up two equations, one for each of the two table t-values. When he solves each of these equations for µ, he will find an interval that he is 95% sure (a statistician would say "with .95 confidence") contains the population mean.

Kevin decides this is the way he will answer the question. His sample contains packs of beers with weights of:

16.25, 15.89, 16.25, 16.35, 15.9, 16.25, 15.85, 16.12, 17.16, 18.17, 14.15, 16.25, 17.025, 16.2, 17.025

He finds his sample mean, x = 16.32 kilograms, and his sample standard deviation (remembering to use the sample formula), s = .87 kilograms. The t-table tells him that .95 of sample t’s with 14 df are between ±2.145. He solves these two equations for μ:

[latex]+2.145 = (36.32 - \mu)/(.87/\sqrt{14})\;\;\;and\;\;\;-2.145 = (36.32 - \mu)/(.87/\sqrt{14})[/latex]

finding μ= 15.82 kilograms and μ= 16.82 kilograms. With these results, Kevin can report that he is "95 per cent sure that the mean weight of a pack of 24 beer bottles is between 15.82 and 16.82 kilograms". Notice that this is different from when he knew more about the population in the previous example.

Summary

A lot of material has been covered in this chapter, and not much of it has been easy. We are getting into real statistics now, and it will require care on your part if you are going to keep making sense of statistics.

The chapter outline is simple:

• Many things are distributed the same way, at least once we’ve standardized the members’ values into z-scores.
• The central limit theorem gives users of statistics a lot of useful information about how the sampling distribution of x is related to the original population of x’s.
• The t-distribution lets us do many of the things the central limit theorem permits, even when the variance of the population, s_x, is not known.

We will soon see that statisticians have learned about other sampling distributions and how to use them to make inferences about populations from samples. It is through these known sampling distributions that most statistics is done. It is these known sampling distributions that give us the link between the sample we have and the population that we want to make an inference about.