Introduction to some ideas of statistics
SampleTesting
Destructive testing is sometimes used to test the ultimate life of a product or design. In the process of testing, the item is destroyed, so valuable information is gained, but obviously not every unit can be tested, so we rely on sample testing. How many units need to be tested, and how frequently during the production cycle, are questions statistics addresses.
Sampling can also be used when it is not feasible, or cost effective, to measure every member of a set. Sampling to figure out the size of fish in a lake is an example. How much sleep does the average U.S. high school student get per night?
Surveys can reveal a lot of useful information, but care has to be taken to make sure that the data is not tainted by any biases. Biases can affect what questions are asked and the order of the questions which can often affect the results. A big question in statistical studies is, who gets surveyed? How the participants are selected can affect the accuracy and outcome of the survey.
Examples of biases are:
Age bias, religious bias, political bias, racial bias, and the list goes on. Biases can affect the outcome of studies and often show up in news reporting of statistical studies, which might be highlighted differently by different parties with different outlooks. Bias can be conscious or unconscious.
Data presentation
Data is often presented in graphical form so that it is easier to see the big picture at a glance. Which graph is best? the scale and which part of the data is to be highlighted are all part of data presentation.
Two variables can be shown to correlate, but that is not the same as showing a causal relationship. Are the two variables changing in sync because one is causing the other or because of some other factor? As we say, “Correlation does not prove causation.”
As a consumer of data, it is always important to try to discern if the data is biased and if it seems reasonable. You may not be able to assess accuracy, but you still could possibly identify biases. It is often helpful to ask “Who paid for the study, and why?”
Mean, mode and median are measures of the central tendency of the data that are typically analyzed. Other areas of analysis such as the measure of standard deviation (the spread of the data around the mean), and outliers are also covered.
Text and Numerical Data
Text data analysis is an important component of general data analysis. Text data can be made up of alphanumeric characters which are treated like characters. Examples of numeric data that are not arithmetic include phone numbers, house numbers, zip codes and course numbers. These data can be converted to numeric data if desired for sorting purposes.
Furthermore, parts of a “text string” can be pulled out of sequence for analysis purposes, or alternatively text strings can be combined. Research the following functions in Excel to see how they are used, and then do the exercise.
Even though there is a very robust Help utility in Excel, one has to be familiar with the language and shorthand used, or the instructions can be unintelligible. Likewise, it is important to know the name of the function you would like to use, or else it can be very time consuming trying to get something done.
Some key ideas and definitions:
The term “text string” is used to designate alpha characters strung together. Alpha characters can be numeric as well as alphabetic, depending how they are formatted. Remember that cell formatting allows you to designate cells as text, numbers (of different types), date (in different styles) etc.
Examples of numeric data that you might want to treat as a text would be zip codes and phone numbers. As a number, the zip code 09999 would not print the leading 0. In order to get Excel (and other databases too) to print the leading zero, the zip code would have to be treated as a text string, or you would have to use “special” formatting.
Logical Test A logical test is a statement that can be evaluated as either True or False, and often is used to evaluate or compare the value of a cell. For example: =A1>10. This logical test will return TRUE if the inequality is true, or FALSE if the inequality is false. In the Help utility, when referring to several logical statements, they will be written as “logical1, logical2….”, which means there are several logical tests, separated by commas.
Logical Statement A logical statement is statement that needs evaluation but will not necessarily return a TRUE or FALSE. See IF command below.
Equal Sign An = sign must be used to tell the program that you are entering something for evaluation such as a formula or a logical statement. In the example above, failure to write =A1>10 would result, not in evaluation of TRUE or FALSE, but rather the writing of the alphanumeric statement A1>10.
LEFT(text, num_char) The LEFT function returns the left-most characters of a string. The first argument gives the location of the string, with the second argument of the command being how many characters you want to capture.
For example, =LEFT(A2,LEN(A2)-1) will return all of the characters of the string in location A2 except the last character since the number of characters captured is equal to the length of the string (LEN( ) ) less one.
RIGHT(text, number of characters) The RIGHT function returns the right-most characters of a string. The first argument gives the location of the string, with the second argument of the command being how many characters you want to capture.
For example, = RIGHT(A2,5) will return the right-most 5 characters from cell A2, and, = RIGHT(A2) will return the one right-most character from cell A2, since, if you do not specify an number of characters, it assumes 1 as default.
MID(text, start_number, number of characters) The MID function returns some specified middle characters of a string in location text, starting with the start_number which is the first character of the MID string, and capturing the number of characters specified. The first argument gives the location of the string, with the second argument of the command indicating where to start the MID string, and the last argument shows how many characters you want to capture.
For example, =MID(A2,3,5) would take 5 characters from the text located in A2, starting with the third character.
Concatenate The CONCATENATE function combines multiple strings into a single string. The syntax is, =CONCATENATE(text1, text2….) where text1, etc. can either be actual text in quotation marks or locations of text. An empty set of quotation marks can be used to create a blank space, i.e., “ “.
For example, =CONCATENATE(A2, B2, C2) would combine the three cells into the one where the concatenate command is entered. =CONCATENATE(“The average mileage for”, “ “,A2, “ “, is, “ “, B2, “ “, “mpg.” The first output might look like: The average mileage for Ford is 32.2 mpg. Where A column would be the list of manufacturers, and column B would be the average mileage.
If The logical test IF, tests the contents of a cell and depending on the results of the test returns different values. This function can be used with numeric or string data.
The syntax of the command is: =IF(logical test, value if true, value if false).
An example could be: =IF(A2=”red”,1,0). This would return 1 if the cell contents = red, and 0 otherwise. This is helpful for counting how many times “red” appears in the examined cells.
Another example could be: IF(A2=25, “red”, “ “), which would enter red, if the content of cell A2 =25, otherwise it would leave the output blank.
And The logical test AND, tests logical statements, and depending on the results of the test returns TRUE or FALSE; TRUE of all logical statements are true, and FALSE otherwise. The syntax is, =AND(logical1, logical2,…) .
For example, =AND(A2>85, A3= “History”) will return TRUE if both are true, otherwise FALSE.
Linear Programming methods enable us to find the optimum, maximum or minimum, solution to an objective function in two variables with several conditions, which may have an infinite number of feasible solutions. The step-by-step method below is to be followed for two variables while the Simplex Method is used when there are more than two variables.
These notes are being posted because it is the desire of Dr. Joel Simon and his family to share these notes with other Professors and Lecturers who could benefit from his work. The notes consist of lecture outlines and class problems with solutions.
Dr. Simon is a well known math and physics teacher who earned his PhD in theoretical physics but found his calling teaching honors math and physics in high school and night college. He also taught graduate courses in higher math and physics.
Dr. Simon was awarded the Engineers Week Council’s Math Teacher of the Year Award, a coveted award granted to one math teacher in the City of Philadelphia each year.
Calc only, no stats — the name of course was “Calc and Stats.”
Calc and Stats Quiz 2 See solutions below:
This test is on calculus only; no stats. It is good practice.
Click below for a study sheet that concisely summarizes the rules that govern how to work with algebraic expressions that have exponents.
Making a Box Plot on the TI 84
It is easier than you think.
The answer keys for tests 2, 3, and 4 are only a click away below. I will post the answers to test 5 after I see that everyone has taken it. Please e-mail me any questions.
Do More Math 