Mean or Average

In statistical applications, the average is almost always called the mean and is defined formally for a sample as:

6.1

where (pronounced x bar) designates the sample mean and ∑ (pronounced sigma) indicates a summation of all values x_i from i = 1 to i = n, with n being the sample size.

For the data shown in Figure 6.1, Equation 6.1 would be executed as (2 + 8 + 11 + ⋯ + 14)/12 (the three dots mean to continue adding the numbers), and the result would be the mean, which is approximately 8.92. In this example, the median is 8 and the mean is 8.92. The mean and the median are frequently different. For example, a large number of people stay in the hospital for short periods of time and incur relatively small charges, whereas a small number of people have relatively long hospital stays and incur high charges. Therefore, for hospital days and cost per admission, the mean is higher than the median, and the distribution of both is skewed to the right. If the mean of a data set is lower than the median, the data are skewed to the left.

“Mean” is also referred to as simple average.

Median

“Median” refers to the middle value in an ordered set of values, or, if there is an even number of values, it refers to the average of the two middle values. If we look at the data from Figure 6.1, as reordered in Figure 6.2, we can see that the sixth and seventh observations (out of 12, an even number) are both 8. The average of two values of 8 is also 8, so that the median for this set of data is 8. Typically, it is more likely that the median will be used to describe a set of data than the mode will be used. It is not uncommon to see the median referenced as the measure of central tendency in such examples as median family income, median age, and median inpatient days. It can be said that half the values of a data set lie above the median and half of the values lie below the median.

“Median” refers to the middle of the data set.

Mode

“Mode” refers to the most commonly appearing value in a set of data. In this case, eight minutes is the most commonly appearing value and is the mode, or modal value. The physician spent eight minutes with each of three patients. Data sets can be bi- or multimodal if more than one value is represented an equal number of times. If, for example, patient 5 had received seven minutes of the physician's time, both eight minutes and seven minutes would have been modal values. In general, although the mode is usually listed as one of the three measures of central tendency, it rarely figures into any further statistical applications.

Mode is the most commonly appearing value in a data set.

The mode provides a measure of central tendency that uses only the most commonly appearing value. The median essentially provides a measure of central tendency that uses only the middle value (for an odd number of observations) or the two middle values (for an even number of observations) in an ordered array. The mean uses all the information in the data set to provide a measure of central tendency. The mean should be a familiar concept to most readers. It is the average. Few people taking their first statistics course are unfamiliar with how to obtain an average for a set of numbers; you just add them and divide by the total number of numbers.

Excel Functions for Central Tendency

Microsoft Excel provides a function for each of the three measures of central tendency. These are =MODE(), =MEDIAN(), and =AVERAGE() (not =MEAN()). The =MEDIAN() and =AVERAGE() functions work just as one would expect, producing the correct value. The =MODE() function will produce the correct value if the data set has only one mode. If it has more than one mode, it will produce a result that indicates only one of the modal values—the value that appears earliest in the data set.

Most of the work we do in this book with measures of central tendency will be done with the mean. The presentation in Equation 6.1 refers specifically to the mean of a sample of observations from some larger population. In the case of the data shown in Figure 6.1 or Figure 6.2, the population would presumably be all patients who ever did or ever will come to the office of the particular physician in question. If we were to consider the mean as a measure of central tendency for the entire population of patients who ever did or ever will visit this physician, it would be designated by slightly different symbols. The mean for a population is defined as shown in Equation 6.2. In Equation 6.2, the sample designation has been replaced by the population designation μ and the sample designation for all the observations in the sample, n, has been replaced by the population designation for all possible observations in the population, N.

6.2

where μ (the small Greek letter μ, pronounced mu) designates the population mean and N represents all patients who ever have or ever will visit the physician. The value of x_i is summed from i = 1 to N.

The point of using the formula in Equation 6.1 to calculate the mean of the sample is that it is what is known as an unbiased estimator of the true population mean defined in Equation 6.2. An unbiased estimator is a sample estimator of a population parameter for which the mean value from all possible samples will be exactly equal to the population value. Thus, since is an unbiased estimator of μ, the mean of the values of from all possible samples taken from the population will be exactly equal to μ. More is said of this in the next section.

Range

A simple measure of dispersion is the range. The range is the difference between the largest and smallest values in the data set. There are also a number of modifications on the range, such as the range within which are found the lowest 25 percent of cases, the highest 25 percent of cases, the middle 50 percent of cases, and so on. The range is a measure of dispersion that is similar in concept to the median as a measure of central tendency. It uses only a limited amount of the data. Except for some descriptive purposes, the range is rarely used in statistical analysis applications and is therefore not discussed further in this book.

Variance and Standard Deviation

The measure of dispersion that, like the mean, uses all the information in the data is the variance, or its square root, the standard deviation. Both the variance and standard deviation are used extensively in statistical applications and are, in fact, the basis upon which much of statistics is constructed. The variance for a sample is defined mathematically as shown in Equation 6.3.

6.3

where s² designates the variance and ∑ indicates a summation of all values of (x − )² from i = 1 to i = n.

The standard deviation for a sample is calculated as shown in Equation 6.4.

6.4

Calculating Variance and Standard Deviation with =VAR() and =STDEV()

Basically, the variance is a measure of the extent to which each observation differs from the mean of all observations. For the data shown in Figure 6.1, the variance calculation is shown in Figure 6.3, a spreadsheet representation of the calculation. In Figure 6.3, the patient number is shown in cells A3 to A14, and the minutes spent with the physician are shown in cells B3 to B14. Cells C3 to C14 show the calculation of the numerator portion of Equation 6.3. The mean value for the data (shown in cell B15) is subtracted from each cell B3 to B14, and the result is then squared (the Excel formula for cell C3, for example, is =(B3-$B$15)ˆ2). The title in cell C2 is a convenient way to show this operation in Excel, because it is difficult to put bars above letters or to create superscripts to show the square process. The ˆ2 is the Excel convention for raising a value to a power. Cell C15 is the sum of the squared differences (=SUM(C3:C14)), and cell C16 is the actual variance as calculated by the formula in Equation 6.3 and Figure 6.3 (=C15/11). The number in cell C17, which is identical to cell C16, is the variance as calculated using the =VAR() function. This function can be seen in the formula bar above the spreadsheet, because cell C17 was the highlighted cell when this image was captured. It is relatively easy to confirm the results shown in Figure 6.3. All that is required is to enter the data in column B into a spreadsheet and follow the formula given in Equation 6.3. Table 6.1 shows the formulas for Figure 6.3.

The variance is a measure of dispersion calculated on the basis of squared differences between each observation and the mean of all observations. This produces a measure of dispersion that is not on the same scale as the measure of central tendency (the mean). The variance actually corresponds more closely to the square of the mean than to the mean itself. To get the measure of dispersion back to the same scale as the mean, the measure of dispersion that is typically used is the standard deviation, shown in Equation 6.4.

The value of the standard deviation for the data shown in Figure 6.3 is approximately 3.48. You can confirm this by using the =STDEV() function, which Excel provides for calculating the standard deviation on the data shown in Figure 6.3. You can also confirm that the =STDEV() function actually produces the square root of the variance by using the =SQRT() function to calculate the square root of the variance given in the same figure. You should find that the results of the two methods are the same.

The formulas in Equations 6.3 and 6.4 provide the variance and standard deviation for samples from a larger population. If we wished to calculate the variance and standard deviation for the entire population of patients who ever had or ever would attend the clinic in question, we would use the formula shown in Equation 6.5. The equation for the standard deviation of the population, designated σ², is just the square root of the value in Equation 6.5.

6.5

where σ² designates the variance and N indicates that the summation of (x_i − μ)² is over all observations in the entire population.

An Example of Sampling with Replacement: In Other Words, Why n − 1?

Let us imagine a very small population that consists of only four observations. These four observations take on the following four values: 2, 4, 5, and 7. If this is the entire population, it is relatively easy to confirm that the mean, μ, for this population is 4.50, using the formula in Equation 6.2, and the variance σ² is 3.25, using the formula in Equation 6.5. To examine the effect of taking all possible samples, we can limit ourselves to all the possible samples of a given size. Whatever the outcome for a sample of any size, the outcome for all samples of other sizes will be the same.

First, what is sampling with replacement? Suppose we want a sample of size two, and we randomly select one observation from our very small population, which turns out to be 5. Now we will draw a second observation from the population to complete the sample of two. In sampling with replacement, we return the observation 5 to the population so that it has an equal chance, along with all other observations, of being drawn in the second selection. If we were sampling without replacement and drew the observation with the value of 5 on the first draw, that observation would not be available for the second draw, which would be limited to the observations with values of 2, 4, and 7.

If we are taking samples with replacement, there are Nⁿ different samples of size n that can be taken from a population of size N. If we are sampling without replacement, there are different samples of size n that can be taken from a population of size N. This is a substantially smaller number than Nⁿ. This is discussed in more detail later, but you should recognize this as being part of the binomial distribution formula. In the case of our population of four observations, there are 4², or 16, different samples of size two that can be taken from the population with replacement. These samples are shown in Figure 6.4.

Using Excel to Demonstrate Why n − 1 Is Used

Cells B2:B5 in Figure 6.4 are the values taken on by each of the four members of the population. The value in cell B8 is the mean of the population calculated with the =AVERAGE(B2:B5) function. The value in cell B9 is the variance of the population calculated with the =VARP(B2:B5) function, which calculates the population variance as given in Equation 6.5. The cells C2:C17 just number the 16 different samples of size two that can be taken from the population of four with replacement. Cells D2:D17 show the first observation in each sample, and cells E2:E17 show the second. Cells F2:F17 represent the means of each sample calculated, using =AVERAGE() for each of the corresponding two cells in D and E. For example, cell F2 is calculated as =AVERAGE(D2:E2). Cell F19 is calculated as =AVERAGE(F2:F17). It can be seen that it matches the value in B8. Similarly, cells G2:G17 are calculated by the =VAR() function, with the appropriate references in columns D and E. Note that the =VAR() function divides by n − 1 rather than by n. The number in cell G19 is calculated as =AVERAGE(G2:G17) and is the average of all the sample variances. It is equal to the number in cell B9. If, instead of using =VAR() to calculate the sample variances in G2:G17, we had used =VARP(), the average of all sample variances (cell G19) would have been not 3.25 but, rather, 1.625—a value much smaller than the true population variance.

This is a demonstration of the appropriateness of using the formulation in Equation 6.3 as the sample variance. This ensures that we obtain a measure that will be an unbiased estimate of the population variance as calculated in Equation 6.5. But two things should be remembered. First, although it is a demonstration (and it works every time), it is not a proof. Also, this relationship between Equation 6.3 and Equation 6.5 holds only for samples taken with replacement. In most cases, samples are not taken with replacement. If you drew a sample of people for interviews, you would not want to interview the same person twice if it happened by chance that he got into the sample twice. When one is sampling without replacement, the formulation in Equation 6.3 is not an unbiased estimate of the population variance. But, in general, the size of a sample relative to the size of a population makes it highly unlikely that any single population item will be selected more than once, so that sampling with replacement might be assumed. Interestingly, when one is sampling without replacement, the average of all sample variances calculated by Equation 6.3 turns out to be numerically equal to the variance of the population calculated by replacing N in Equation 6.5 with N − 1.

One other point should be made in regard to unbiased estimates of dispersion. The typical measure of dispersion used is the standard deviation or the square root of the variance. The average value of the standard deviation of all possible samples from any population will not be equal to the standard deviation of the total population, calculated by any means. This is because, in general:

Cumulative Differences

One such possibility might be the sum of the differences between each value and the mean of all values, which could be shown as given in Equation 6.6.

6.6

Unfortunately, this strategy does not work well. Because the mean is the numerical midpoint of the distribution, the sum of all the negative numbers will always equal the sum of all the positive numbers derived from the operation given in Equation 6.6. In turn, the result of this operation is always zero for every data set. Because of this, Equation 6.6 cannot serve as a useful measure of dispersion.

Absolute Differences

But what if a measure of dispersion were the absolute value of the difference between each observation and the mean of all observations? Such a possibility would be expressed as Equation 6.7.

6.7

Unfortunately, this criterion also does not work well for a measure of dispersion if our measure of central tendency is the mean. One assumption we are making is that any measure of dispersion we ultimately choose should be centered on the mean. This assumption is true for both Equations 6.6 and 6.7.

Measures of Dispersion and Being Centered at the Mean

To understand the notion of centered in the context of Equation 6.7, consider Figure 6.5. This table shows a column (cells A16:A22) of seven data points that take on the values 1 through 7, respectively. It is easy to confirm that the mean of these seven items is 4. Cells B15:H15 contain seven possible constants that may be substituted for the mean in Equation 6.7, including the true mean of 4. The formula line in Figure 6.5 shows that the value in cell B16 is the absolute difference between those in cells A16 and B15. Cell B17 is the absolute difference between cells A17 and B15. The remaining values are calculated in the same way. The sums of the absolute differences in each column are given in cells B23:H23. The value 12 in cell E23, which corresponds to the mean, is the smallest of the sum of absolute differences. In this sense, the result of Equation 6.7 is centered on the mean in this data set. If any number other than the mean is used as the constant subtractor, the result of the sum of absolute differences will be greater than the value when the mean is employed. In general, a working assumption is that whatever measure of dispersion is chosen, it should be centered on the mean. By being centered at the mean, the measure of dispersion then produces a minimum sum of differences at, and only at, the mean.

But Equation 6.7 does not always produce the minimum value at the mean. In fact, it turns out that Equation 6.7 always produces a minimum value at the median. To see an illustration of this, consider the data shown in Figure 6.6. This table again shows seven observations (cells A30:A36). In this case, it is not difficult to confirm that the mean of the data elements is 4, whereas the median is 2. Again, the values in cells B30:H36 are the absolute differences between each data point and the constant values listed in cells B29:H29. Now it can be seen that the minimum value of the absolute differences (16 in column C) is centered not on the mean of 4 but, rather, on the median of 2. This is the primary reason that the absolute differences between each value and the mean are not generally used as a measure of dispersion when the mean is the measure of central tendency. (It can also be shown that for data sets that have even numbers of cases, the absolute difference is centered in the sense discussed here on both values that define the median and on all possible numerical values between them.)

Now what of the sum of squared differences (Equation 6.3) as a measure of dispersion? Does it provide a measure that is centered on the mean? Consider Figure 6.7, which shows the sum of squared differences for the data displayed in Figure 6.6. Now, the minimum value of the sum of differences (squared) is at the mean of 4, so the sum of squared differences is centered at the mean. The reader might wish to confirm that for the data in Figure 6.5, the sum of squared differences is a minimum at the mean. It will turn out that for any data set, the sum of squared differences has the desirable property that it is centered at the mean in the sense that the sum of squared differences between the data elements and any constant will be a minimum when the true mean is the constant.

Using the Mean and Standard Deviation to Create a Frequency Distribution

The use of the mean and standard deviation in creating a frequency distribution is shown in Figure 6.8. In Figure 6.8, column B contains the countries of the world ranked by HDI number. There are actually 162 countries ranked in column B, but only the first 13 are shown in the figure. Column C contains the HDI score for the top 13 countries. The maximum score (for Norway) is 0.939. The minimum score (which is not shown and is for Sierra Leone) is 0.258. The mean of the HDI score across countries is shown in cell E2 as 0.684, and the standard deviation is shown in cell E3 as 0.181. Beginning in cell E6 is the sequence of numbers −3, −2, …3. This sequence represents a distance from the mean in numbers of standard deviations ranging from 23 to 3. These distances are actually calculated in cells F6:F12. The equation that is used to calculate the distances is shown for F6 in the formula line as =$E$21E6*$E$3. This formula notation means that when the formula is copied from cell F6 to cells F7 through F12, the formula will refer to the relevant consecutive cells from E7 to E12 but will always refer to cell E2 (the mean) and cell E3 (the standard deviation).

Table 6.2 shows the formulas for Figure 6.8.

Creating a Frequency Distribution Using the =FREQUENCY() Function

The frequency distribution is shown in G6:G12; it was calculated using the =FREQUENCY(C2:C163,F6:F12) function. You will recall that it is always necessary to use Ctrl+Shift+Enter to get the =FREQUENCY() function to work correctly because it is one of the functions that compute a value for more than one cell at a time. Two things might be noted about the frequency distribution. First, the sum of the frequency distribution given in cell G15 as 162 (=SUM(G6:G14)) confirms that all of the countries are included in the frequency distribution. Second, the HDI value for every country is contained in the range from 3 standard deviations below the mean to 2 standard deviations above the mean. (Remember that the Bin value refers to the top of a particular range so that 22 contains all countries with HDI scores between 0.140 and 0.321.)

The histogram created by this frequency distribution of the HDI data is also of interest; it is shown in Figure 6.9. Several things might be mentioned in regard to this figure. First, the histogram includes data only for the range from −3 to +3 standard deviations. Typically, most of the observations in almost any data set will be in the ±3 standard deviation range. A second point of interest is that column I contains Bin labels that were not used in the calculations in any way but were used to label the horizontal dimension of the graph. This was discussed in Chapter 4 in the section on construction of graphs. It should be remembered that the Bin values actually used in the =FREQUENCY() formula represent the top of the BIN range. For example, the −2 in cell E7, which is associated with the value 0.321 in cell F7, actually refers to the range from −3 to −2 standard deviations. Column H represents the percentage of countries in each BIN range.

The histogram itself shows a distribution of the frequencies that is not atypical for many distributions. The observations tend to cluster near the mean and be fewer further from the mean. Ninety-five of the 162 countries (58 percent) have HDI scores within 1 standard deviation of the mean. When one counts the number of countries within 2 standard deviations of the mean, there are 157 of the 162 (97 percent). This type of distribution of individual cases around the mean is typical of many distributions found in the real world. In many cases, data are distributed approximately in what is known as a normal distribution. The normal distribution was introduced in Chapter 5. There, it was indicated that the normal distribution is symmetrical with an equal number of observations above and below the mean. A second characteristic of the normal distribution is that approximately 68 percent of the observations are within ±1 standard deviation of the mean and 95 percent of the observations are within ±2 standard deviations. The distribution in Figure 6.9 is not actually symmetrical (70 countries are in the range of 1 standard deviation above the mean, whereas only 25 are within the range of 1 standard deviation below). Still, the percentages of 58 percent at ±1 standard deviation and 97 percent at ±2 standard deviations are not far from the normal distribution percentages of 68 percent and 95 percent.

Constructing a Normal Distribution: The =NORMDIST() Function

The way in which the normal distribution in Figure 6.10 was constructed in Excel tells us a lot about the kinds of information that can be obtained from Excel. Figure 6.11 shows a portion of the spreadsheet that was used to produce the normal distribution shown in Figure 6.10. Column A, labeled StDev, represents the distance from the mean in standard deviation units. Cell A2 is shown as −4, which means that the cell represents 4 standard deviation units below the mean. Cell A3, with the value −3.9, represents 3.9 units below the mean, and so on. The values in column A increase to 4 in cell A82 (not visible in the figure). Column B is labeled Normal Dist. The value in cell B2 is produced by the =NORMDIST() function and represents the proportion of the normal distribution that falls at the point represented by −4. The proportion of the normal distribution falling at any single point is given in Equation 6.8. The values in column C, labeled Normal Dist2, represent the normal distribution values calculated by the formula shown in Equation 6.8. The way in which the formula is expressed in Excel is shown in the formula line in Figure 6.11.

6.8

The syntax of =NORMDIST(), which is not shown in Figure 6.11, is =NORMDIST(Cell Ref,0,1,0). The =NORMDIST() function takes four arguments: (1) the cell reference (Cell Ref) for which the point proportion of the normal distribution is desired (e.g., 24, 23.9, etc.), (2) the mean of the distribution (in this case, 0), (3) the standard deviation of the distribution (here given as 1), and (4) a value of 0, indicating that the distribution desired is the normal probability density function and not the cumulative normal distribution. The cumulative normal distribution will be discussed later.

Characteristics of the Normal Distribution

Looking at Figure 6.10, it should be recognized that there is a very small proportion of the observations in a normally distributed data set that lies outside the range −3 to +3 standard deviations. In fact, a normal distribution has about 99 percent of its observations in the −3 to +3 range on either side of the mean. However, some proportion of a normal distribution (albeit a very small proportion) is out beyond 4 standard deviations, 5 standard deviations, and even 20 standard deviations. The probability of its being out at those extremes is finite, but very, very small. Although it is not obvious from Figure 6.10, about 68 percent of the observations lie in the range from −1 to +1 standard deviations and about 95 percent lie in the range from −2 to +2 standard deviations.

It is basically correct to say that 68 percent of all observations in a normal distribution are within 1 standard deviation on either side of the mean value. It is also true that approximately 95 percent of all observations are within ±2 standard deviations of the mean. In addition, around 99.7 percent of all observations lie within ±3 standard deviations of the mean. But an interesting characteristic of normal distributions is that they are based on scales that are continuous rather than discrete. For example, if one were measuring the height of U.S. adult men, the recorded measurement would be limited by the measuring device itself. A standard ruler probably has no units smaller than quarters of an inch. Therefore, one could measure a person's height to the nearest quarter inch only.

For a normal distribution it can be said that around:

• 68 percent of all observations are within +1/−1 standard deviation of the mean.
• 95 percent of all observations are within +2/−2 standard deviations of the mean.
• 99.7 percent of all observations are within +3/−3 standard deviations of the mean.

However, a more sophisticated measuring technique might make it possible to measure with more precision. In fact, it might be discovered that someone who was measured as being 5 feet, 10 and 1/2 inches tall might actually be 5 feet, 10 and 7/16th inches tall. A still more sophisticated measurement tool might measure that person as actually 5 feet, 10 and 455/1,000th inches tall.

By continuing to increase the precision of the measurement, we are able to measure things on what might be considered a true scale of infinite possibilities A consequence of this theoretical ability is that the probability of any given observation at any one specific spot in the normal distribution (e.g., exactly 1 standard deviation above the mean) is actually zero. We then have a finite number of objects distributed over an infinite number of sites (points in the scale). In turn, the probability that any one object will fall on any specific site will be the number of observations divided by infinity, which will always come out to zero. The consequence of this is that the actual proportion of observations between two points in a normal distribution (say, 2 standard deviations below to 2 standard deviations above the mean) cannot be calculated from the formula in Equation 6.8. Instead, the actual proportion of observations within any interval in the normal distribution must be taken from the cumulative normal distribution. This distribution is discussed in detail in the following subsection.

Approximating the Area under a Normal Curve

The actual value we obtain is the integral of the area under the curve from −∞ to the number in question, but it can be approximated using one of several methods. The formula in Equation 6.9 is the way the area under the cumulative normal distribution is calculated by Excel, and it duplicates the result of the Excel =NORMDIST(x,mean, stdev,1) function. This formula is an approximation, rather than an exact solution, but its results are far more readily obtained than any exact solution.

6.9

where

Equation 6.9 works for values of x greater than 0. If the value of x is less than 0, it is necessary to subtract p(x) from 1 in order to get the probability from ∞ to the value of x less than 0. For example, to obtain the cumulative proportion at −1, as shown in Figure 6.12 (approximately 15 percent, by the graph), it would be necessary to find the cumulative proportion at the value of 1, as shown in Figure 6.12, and subtract that value from 1 to get the cumulative proportion at −1.

There is no particular reason why you, as a user of this book, will need to know the formulas in Equation 6.8 or those in Equation 6.9. But we are always interested in how things happen. If Excel provides a value for the cumulative normal distribution, we like to know, if possible, where Excel got the value. Thus the equations are given. You may use them as you wish.

Understanding the Cumulative Normal Distribution in Excel

Figure 6.13 shows the values generated by a cumulative normal distribution based on the assumption that U.S. newborns average 3,600 grams and that the distribution of the weight of newborns has a standard deviation of 850 grams. Column A shows 0.5 standard deviation unit from the mean of 3,600. Column B shows the weight in grams that corresponds to each standard deviation unit from the mean, assuming a mean of 3,600 grams and a standard deviation of 850 grams. Column C is the probability that was obtained, as the formula line indicates, with the =NORMDIST() function. The first argument is the weight given in column B, the second is fixed as the mean given in cell E3, the third is the standard deviation given in cell E4, and the final argument is the 1 denoting cumulative distribution.

To understand the logic and use of the cumulative normal distribution, suppose there is an interest in knowing what proportion of newborns will weigh less than 2,000 grams (2 standard deviations below the mean). To determine this, given the assumption that the average is 3,600 grams and the standard deviation is 850 grams, it is necessary only to look at the probability given in cell C6 in Figure 6.13. According to that cell, fewer than 3 percent of newborns would weigh as little as 2,000 grams. What if the interest were in knowing what proportion of newborns would weigh more than some amount at birth—say, 4,875 grams? Still, assuming a mean of 3,600 and a standard deviation of 850, it is possible to see that the probability associated with 4,875 grams (cell C13) is approximately 0.93. But this is the proportion of newborns who will weigh 4,875 or less. To get the proportion that will weigh 4,875 or more, it is necessary to subtract the proportion at 4,875 from 1; so, the actual proportion who will weigh more than 4,875 grams at birth is 1 − 0.93, or approximately 0.07.

Finding the probability of any occurrence that can be described by a normal distribution with a known mean and standard deviation can be accomplished with the use of the =NORMDIST() function. Suppose the weight of 13-year-old girls is normally distributed with a mean of 100 pounds and a standard deviation of 15 pounds. What is the likelihood of finding a 13-year-old girl who weighs less than 70 pounds? The answer can be found in Excel with =NORMDIST(70,100,15,1), which returns the result of about 0.02. Suppose the body temperature of healthy adults is normally distributed with a mean of 98.6 degrees Fahrenheit and a standard deviation of 0.5 degrees. What is the probability of observing a healthy adult with a temperature of 101 degrees Fahrenheit or more? The answer can be found using Excel's =NORMDIST(101,98.6,0.5,1) function, which produces 0.9999 to four decimals. We do not want to know the probability of finding a healthy adult with a temperature below 101. Rather, we want the probability of finding a healthy adult with a temperature of 101 or above. That would be found as 1 − 0.9999, or 0.0001 at four decimal places. Because the probability of finding a healthy adult with a temperature of 101 is so small, we would typically conclude that such a person could not be healthy.

Calculating and Interpreting the Means of the Samples

Because we now have 250 sample means, we decide that the value we will accept for the true mean of the population will be the mean value of the 250 samples. When we calculate the mean of the 250 sample means, we discover that it is 5.22. So 5.22 becomes the estimate to represent the true mean of the population. But that is not the end of the story. Clearly, the mean values from our 250 samples have a distribution of their own. What can we say about that distribution? To look at that distribution, we are going to graph it in standard deviation units from the mean. But we are not going to use population, or even sample standard deviation units. We are going to use the standard deviation units of the sample means.

The standard deviation of the sample means is 0.402. The largest sample mean in the set of 250 was 6.45, and the smallest was 4.29. Given the standard deviation of 0.402, we would expect that about 95 percent of the sample means would be between 4.42 and 6.02, which is 2 standard deviations on either side of the mean of the samples, if the 250 means are normally distributed. The distribution of the 250 sample means is shown in Figure 6.15.

Figure 6.15 shows a very different picture when compared with Figure 6.14. Although the actual discharges as shown in Figure 6.14 are heavily skewed to the right, the means of 250 samples selected from that original data, shown in Figure 6.15, are almost normal in their distribution. It is not totally obvious from Figure 6.15, but about 70 percent of all sample means are within one standard error of the mean of all samples (the normal distribution has 68 percent). Also, about 97 percent of the mean values are within 2 standard deviations of the mean of all samples (the normal distribution has 95 percent). So the distribution shown in Figure 6.15 is seen to be a fairly close approximation of a normal distribution.

This finding that the means of 250 samples approximate a normal distribution reveals an interesting characteristic of sampling—or, rather, the means of samples. When the sample size is relatively large (30 or more in the sample is usually considered relatively large), the distribution of sample means tends to be normal, regardless of the nature of the distribution of the actual values. This characteristic of the distribution of sample means will be very useful when the notion of confidence limits is examined in Chapter 7.

The Concept of Standard Error

In an examination of the distribution of sample means, a relatively new notion was introduced. This is the standard deviation of the sample means. Up until this time, the discussion of standard deviations has focused on the individual observations. When the standard deviation of actual lengths of stay was calculated for the hospital data, it was 4.12 days. But when the standard deviation of the 250 sample means was calculated, it turned out to be only 0.402 days. A question that might now be asked is whether there is any defined relationship between the standard deviation of the individual observations and the standard deviation of the sample means. This leads to the introduction of the concept of the standard error. The standard error is the standard deviation of the sample means for samples of a given size.

The standard error is the standard deviation of the sample means for samples of a given size.

If the true population variance is available, the standard error is given as shown in Equation 6.10. This shows that there is a very specific relationship between the standard deviation of the population and the standard error of the means of samples drawn from the population. This relationship is precisely that the standard error of the means is the standard deviation of the individual observations divided by the square root of the sample size.

6.10

The formula in Equation 6.10 provides an interesting comparison. However, let's go back for a moment to variance rather than the standard deviation. From Figure 6.4 it can be seen that the mean of the variance of all possible samples of size two from a population of four will be exactly equal to the variance of the four actual observations. Similarly, the standard deviation of all four observations divided by the square root of the sample size of two will be exactly the same as the standard deviation of the means from all possible samples of size two taken from the population of four. This is demonstrated in Figure 6.16, which is the same as Figure 6.4. However, Figure 6.16 now shows the population standard deviation divided by the square root of the sample size (cell B10) and the standard deviation of the means of all samples of size two (cell F20). These two values are exactly the same. (It should be noted that the standard deviation of the means of all samples is calculated using the population standard deviation function =STDEVP(). This is appropriate, because this set of means is the population of all mean values.)

Now let us return to our 250 samples of size 100. The standard error of the mean for samples of size 100 using Equation 6.10 is 4.12 divided by the square root of 100, or 0.412. This is close to the standard deviation of the means that was calculated from the 250 samples we selected (0.402) but is not identical. Why the difference? It is simply because we do not have all the possible samples of size 100 that could be selected from the 12,000 discharges. If we sampled with replacement, there would be 12,000¹⁰⁰ such samples—a very large number indeed. However, if we did have the means of all those samples, they would be equal to the standard error as calculated in Equation 6.10, or exactly 0.412.

Drawing Samples Using Excel's Data Analysis Add-In

We will draw a large number of samples of size 30 from this population using the Data Analysis add-in. The Data Analysis add-in is found in the Data ribbon in the Analysis tab. Selecting the Data Analysis menu in the Analysis tab will invoke the Data Analysis dialog box shown in Figure 6.18. In the Data Analysis dialog box we select Random Number Generation.

Selecting Random Number Generation will invoke the Random Number Generation dialog box, shown in Figure 6.19. This dialog box (shown along with the part of the spreadsheet that contains prenatal visits and the proportion of women making those visits) contains several items. The first category of information that needs to be supplied is the number of samples desired, here labeled Number of Variables. We are going to generate 100 separate random samples, so we enter 100 in the Number of Variables field. Each sample we generate will include 30 observations, so we enter 30 in the Number of Random Numbers field. Figure 6.19 shows a dotted line from cell A2 to cell B16, the range that is put into the value and probability input range. Column A represents the discrete visits (from 1 to 15), and column B represents the probability of any woman making each number of visits during her pregnancy. In the Random Seed field we enter 3,275, which was generated using the =RAND() function, dropping the decimal point and taking only the first four digits. The last piece of information required by the Random Number Generation dialog box is where the output will go. We want it on a new sheet, so we click the New Worksheet Ply radio button.

Calculating the Measures of Central Tendency and Dispersion for the Sample

When we click OK in the Random Number Generation dialog box, Excel generates 100 samples of prenatal visits of size 30, based on the probabilities given in column B. The result of the generation of these 100 samples is shown in Figure 6.20. This figure shows only a portion of the spreadsheet in which the random numbers have been generated. Each column in the spreadsheet shown in Figure 6.20 represents a separate sample. The numbers in each column represent the number of visits made by the women selected to be part of each sample of 30. The first woman selected as part of the first sample (cell A1) made three visits to the clinic. The second woman selected for the first sample made four visits to the clinic, and so on. Because there are 30 observations in each sample, the numbers in each column actually run to row 30. Because there are 100 samples, the numbers in the columns run to column CV.

Given these 100 samples of size 30, we can calculate the mean, standard deviation, and standard error of each sample, as shown in Figure 6.21. In this figure, a first column has been added to the spreadsheet to provide a place to put the labels for mean, standard deviation, and standard error. This figure shows the last six numbers selected in the first six samples, and row 32 displays the mean of the number of visits for the 30 women in each sample. Row 33 shows the standard deviation as estimated by this sample, and row 34 shows the standard error. The calculation for the standard error for column B—the first sample—is shown in the formula bar.

Let us now look at the distribution of means from a large number of samples. Figure 6.22 shows the distribution of sample means from 250 samples of size 100 taken from the population of women who have come for prenatal visits. The x-axis represents standard deviation units from the mean. The designation −2 shows those samples with mean values from −3 to −2 standard deviations from the mean. The designation −1 shows those samples with mean values from −2 to −1 standard deviations from the mean, and so on. It is clear from this figure that sample means from relatively large samples approximate a normal distribution, even though the original data may not.

This section discusses the use of Excel to generate a large number of samples with given attributes. It is important to remember, however, that, in general, one never draws more than one sample. The discussion here is presented solely for the purpose of helping the reader understand the use of the discrete sampling option in the random number generation add-in. This example is referred to again in Chapter 7, which discusses the calculation of confidence limits.

What Is the t Distribution?

Whereas the normal distribution assumes an infinite number of observations, the t distribution assumes a finite number of observations. Because the normal distribution assumes an infinite number of observations, there is a single normal distribution. Because the t distribution depends on the number of observations in question, there is a whole family of t distributions. Each t distribution corresponds to a separate number of observations—or, more precisely, to a separate degree of freedom.

The t distribution assumes a finite number of observations, whereas the normal distribution assumes an infinite number of observations.

The t Distribution Approximates the Normal Distribution

Let us go back now to the t statistic. The t distribution, which approximates the normal distribution, depends on the degrees of freedom. A t distribution with 100 degrees of freedom (a sample of size 101) will be more nearly normal in shape than a distribution with 4 degrees of freedom (a sample of size 5). Figure 6.29 shows two different t distributions. The white distribution is the t distribution for degrees of freedom equal to 100. The dark gray distribution is the distribution for degrees of freedom equal to four. The horizontal axis shows standard deviations from the mean.

The distribution for 100 degrees of freedom is nearly normal. The proportion of cases within 1 standard deviation on either side of the mean is 0.68, and the proportion within 2 standard deviations is 0.95. The distribution for degrees of freedom equal to four appears approximately normal (and it is), but as can be seen from the figure, there are fewer observations in the middle of the distribution and more at the extremes. With the distribution for four degrees of freedom, only about 63 percent of the distribution is within 1 standard deviation of the mean and about 88 percent is within ±2 standard deviations.

Using the =TDIST Function

Excel eliminates the need for such tables. Take a look at the two distributions shown in Figure 6.29. We wish to know the exact probability of being outside the range of 2 standard deviations from the mean with 100 degrees of freedom (the white distribution) and with 4 degrees of freedom (the dark gray distribution). The answer can be found with the =TDIST() function. Figure 6.30 shows the exact probability of being outside one, two, and three standard deviations of the mean for 100 degrees of freedom and for 4 degrees of freedom. The formula bar shows the =TDIST() function, which takes three arguments. The first is the number of standard deviations from the mean (column A). The second argument is the number of degrees of freedom (given in B2 and C2). The third argument is the number of tails to consider in the distribution. The number of tails can be 1 if the exact probability of being either above or below a particular standard deviation number is desired (i.e., in only one tail of the distribution). The number of tails will be two if the exact probability of being both above or below a certain number of standard deviations is desired (i.e., in both tails of the distribution). Rows 4, 5, and 6 show the two-tail probabilities, whereas rows 8, 9, and 10 show the one-tail probabilities. As Figure 6.30 demonstrates, for example, only about 5 percent of the distribution with 100 degrees of freedom is beyond 2 standard deviations (two-tail), whereas nearly 12 percent of the distribution is beyond 2 standard deviations (two-tail) with 4 degrees of freedom.

Using the =TINV Function

Excel also provides another useful function related to the t distribution, the =TINV() function. The =TINV() function will return the exact value of t (i.e., the number of standard deviations from the mean) for any probability. So, for example, if one wishes to know the number of standard deviations, it is necessary to go out from the mean in order to include 95 percent of all observations in a t distribution with 100 degrees of freedom; that number can be found with =TINV(0.05,100). The result of the =TINV() function, written as in the last sentence, will be 1.98. This means that it is necessary to go out 1.98 standard deviations on either side of the mean to find the interval within which 95 percent of the observations will fall.

One important point about the =TINV() function is that it always returns the two-tail distance from the mean in standard deviation for any probability. For example, if one wanted the one-tail probability distance for a probability of 0.05 and degrees of freedom equal to 100, it would be necessary to use the function =TINV(0.1,100).