Graphic depiction of sample data points relative to mean
How Do We Compute xBar & s? What are they???

St. Paul's Statistics Introduction: Chapter 5

Calculating Mean and Standard Deviation from Data Sets

Section Goals:

Introduction: In chapter 4, we worked many problems with the Gaussian Distribution and they always required us to know the mean μ (average) and standard deviation σ for the population of 'things' we were analysing. Chapter 4 identified three sources of those values:

  1. Very well established theories, such as those in Physics.
  2. Copious amounts of data (that's how the Physicists got their values)
  3. Values calculated from data samples. When we use this approach, we refer to the sample mean (xBar) and the sample standard deviation (s). We do not use the symbols μ and σ when the values are computed from small to medium data sets.
In this chapter, you will learn how to compute values xBar and s (where xBar is an imposter who pretends to be μ. And s is an imposter who pretends to be σ). We are forced to use imposters xBar and s when we can't get the 'real thing'. In our calculations, we can use 'the imposters' in place of μ and σ.

Computing xΒar (imposter for μ): Please recall that μ is the average value for every item belonging to the population. If we are talking about the length of large construction nails, then μ is the average of length for every 16d size nail produced from Columbus up through the 22nd century. We would measure and add up all the lengths, and then divide the answer by the number of nails.

If we are talking about the weight of Cows, then μ is the average weight for every cow from the time of Moses up through the 22nd century. We would add up all the weights, then divide by the number of cows. We would need a time machine go back and weigh every cow because Moses lived a very long time ago. For most populations, it simply is not possible to get population data to compute μ. That is the reason we have to work with sample values xBar and σ.

So, how do you find the mean of x? Just add up all the x values, and then divide by their number. If you use 'all of them', then you get μ. If you use a sample of them, you will get xBar.

Computing Sample Mean xBar for Peanut Butter: Table 1 below contains price data for various brands of peanut butter sold in a one pound jar. The brands marked *, are fictional and were added to more effectively illustrate concepts. All other data was obtained from U.S. government data.

Peanut butter price data
Brands & Prices for Peanut Butter
Chapter 5 Table 1

We can compute the sample mean 'xBar' based on the 19 prices in Table 1. Likely, you know the method, but here is the example:

xBar = Σpricei / count = (2.50+12.49+2.50+3.00+4.00+ …….. $3.00+$4.00)/19 = $4.22

Presenting Data - A Better Way: From Table 1, it is not obvious, but prices range from $1.50 to $12.99 for a jar of peanut butter. It is not obvious, but Table 1 has 6 products priced at $2.50.

Sort Table 1 from the cheapest to most expensive. The price trends will become obvious. Repeated entries likewise become obvious and we can compute 6 x $2.5 rather than pressing the plus sign repeatedly. The sorted version of Table 1 is shown below:

Peanut butter prices sorted
Peanut Butter - Sorted by Price
Chapter 5 Table 2

Further simplifications are possible. To compute xBar (the imposter for μ), we don't need the names of the peanut butter and we don't need jar size. Repeat entries are handled as shown in Table 3 below:

Peanut Butter prices grouped by price
Price Data Sample Tabulated by Count
Chapter 5 Table 3

An Easier Way to Compute xBar: Based on Table 3, we can compute xBar either by calculator or by computer spreadsheet. We simply multiply 'Count' by 'Price' for each row in Table 3. Typically, these values are inserted into a new column to the right. Then, we 'total down' the new column. Finally, divide by the total by count of data items (19 in this example). Confused? We are just adding up all the Price numbers of Table 2 in a smarter way!

For a table with 19 data entries, it is not very important how you do it. But in the real world, statistics problems often have 150 data rows. For situations like this, it is impossible to look at the data table and draw any kind of conclusion. The data table must be sorted and repeat entries should be documented in a 'Count' column (like Table 3). This approach also makes computation of xBar much faster.

Reduced to math notation, this 'easier way' to compute sample mean looks like this:

xBar = ( Σall i (counti * pricei ) ) / Total Number

In spreadsheet form, the solution looks like this:
Computing mean of peanut butter prices
Spreadsheet Form of Simplified Mean Computation
Chapter 5 Table 4

Homework Problems Chapter 5 :
For problems 1 through 4, use the data set to do the following. You may use either calculator or computer spreadsheet. Show all work.

Chapter 5 Problem 1:
A Washington 'think tank' is encouraging larger budgets for health research. They want to statistically describe U.S. child deaths from Flu. Data from National Institute of Health follows

Paediatric Flu death by year
Child Deaths from Flu
Original Source: National Institute of Health
Chapter 5 Table 5

Chapter 5 Problem 2:
A civil engineering firm is constructing a dam. The concrete has a minimum strength requirement. Test specimens from the first 17 concrete trucks were made and the strength test results are shown below. Follow the instructions and compute mean sample strength. (Data was obtained from source w/o copy-right notice.)

Concrete Strength variation between samples
Concrete Strength Test Results
Chapter 5 Table 5

Banded Data: Very often, data is tabulated as huge lists of items with one entry per row. For example: Paul bought a pair of size 9 shoes → that appears as one data row. Tommy bought a pair of size 9 shoes → that is a different data row. When data items arrive as separate line items, but clearly many of the entries are similar, it makes sense to sort the data, and then group all the size 9 purchases together. It is very common to arrange data into 'bands' that fall into certain value ranges. For shoes, it seems very natural; but 'banded' data is quite common even when the 'x' variable seems continuous and repeats in the data are not evident. We will explore this topic when we study histograms in Chapter 6.

For problems 3 and 4, the data has already been sorted and grouped for you. Ignore the columns of data you don't need. Compute the mean of the data set.

Chapter 5 Problem 3:
Shoe manufacturers are very interested in knowing the percentage of each shoe size sold so they can manufacture shoes in the proportions demanded by the market. Below, is a sorted data set that shows a sample of women's shoe purchases. Find the average.

Distribution of woman shoe size
Woman's Shoe Sales by SIZE
Chapter 5 Table 7

Chapter 5 Problem 4:
Once again, we look at shoe purchases, but this time it is men's shoes. Below, is a sorted data set that shows a sample of men's shoe purchases. Find the average.

distribution of men shoe sizes
Man's Shoe Sales by Size
Chapter 5 Table 8

How is the Mean μ Related to the PDF? I offer a few PDF pictures with the mean shown on the graph. The mean of the PDF graph is always the balancing point! These four images illustrate that idea.

Possible PDF shape (Gaussian)
A Gaussian PDF Mean
Chapter 5 Figure 1

Possible PDF shape (triangular)
A Symmetric Triangular PDF Mean
Chapter 5 Figure 2

Possible PDF shape (ramped)
A NonSymmetric Triangular PDF
Chapter 5 Figure 3

Possible PDF shape (possibly Weibull)
NonSymetric PDF (Possibly Weibull)
Chapter 5 Figure 4

Conceptually, the mean μ will always be at the balance point of the PDF. For right-left symmetric PDFs, the mean will always be in the middle (as is the balance point). When doing math, we don't want to use 'eye-ball' estimates of the mean. We want accurate calculations based on samples. But it is useful for you to be able to look at a PDF and compare it with your calculated answer. Then you confidently judge whether or not your answer looks right.

Mid Chapter Summary:

The Variance σ and imposter s:

We know from Chapter 2 that the standard deviation σ determines 'how wide' the Gaussian PDF is. Chapter 2 Figure 3 is repeated below to refresh your memory. From the figure below, you should get the idea that:

Bigger Standard Deviation σ = Wider PDF Curve

variation of PDF width with standard deviation
3 Different GAUSS PDFs with Difference Variance
Ch 5 Figure 5
Note: The three Gauss PDFs shown above are skinny, to wide because the standard deviation σ is small to large. The curves are shifted to the right for a different reason – because the mean value is different. Next, we will learn how to compute a sample standard deviation from a set of data.

Computing s (imposter for standard deviation σ): In the real world, you usually won't know the standard deviation σ. You will have to compute an estimate based on a sample of data. A sample based variance is usually denoted 's' which is a way of reminding us that it is not the true, population value σ. s is an imposter that pretends to be the variance σ and we can use it in calculations in place of σ. In mathematical notation, the following equation defines how the sample variance s is computed:

Equation for Standard Deviation computation

Steps to Compute Standard Deviation s:
Read through the steps, and study how they accord with the equation above:
  1. Obtain a sample of data. It might be a list showing 9 cows. For each cow, a property of interest (like milk per day) will be documented. e.g. cow No 1 → 8.2 gal etc. for every cow
  2. Compute the sample average value xBar using methods presented earlier in this chapter
  3. For each data item (i.e. for each value xi) , we subtract xBar from it. And square the result.
  4. Next, add up all the squared results
  5. Divide by the number of data entries (n)
  6. Take the square root
Tabular computation of standard deviation
Calculating Sample Standard Deviation
Chapter 5, Figure 5

Homework Set 2 Problems Chapter 5:
Problems 5, 6, 7, & 8: Computation of Sample Standard Deviation s:

For each of the data sets of Problems 1,2,3 & 4, Compute the sample mean and sample standard deviation using the method shown above. You may use either calculator and paper, or spreadsheet.

Turn in the results to your instructor.

End of Chapter 5

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Paul F. Watson

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