Elementary Statistics MATH 112
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This 2 page Class Notes was uploaded by Maybell Larson on Monday October 12, 2015. The Class Notes belongs to MATH 112 at Guilford College taught by Elwood Parker in Fall. Since its upload, it has received 20 views. For similar materials see /class/222112/math-112-guilford-college in Mathematics (M) at Guilford College.
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Date Created: 10/12/15
as Z instructs namely all the values X of the variable X Q27 Describe what instructions each of the following gives szx fr 2xx X2 2xx xn 2xx fgxz 2xx x3 Z xn2 Which computes it Which computes an average Understanding the symbolic language of mathematics is crucial The notes which follow are included to make students very conscious of that language and how easily it can be misused and misinterpreted Footnotes 3 it and 25 already encountered are about mathematical language Advice if it is not clear what is being conveyed in mathematical symbols used ask for clari cation 1 Care must be taken in the way in which a symbol is used the point of the two possibilities repeated values or not for a variable X included above If it is not clear what is meant always ask There is nothing mandatory about naming a variable X any other symbol letter or not is just as valid But by longstanding convention letters at the end of the alphabet are used for variables Also conventionally letters at the beginning of the alphabet are used for constants standing for numbers which can have only one xed value that is constants do not vary Both conventions are followed in these notes 2 It might be tempting to think of the f used above as notation for function But it is not i stands for frequency The point is that the same symbol may change meaning depending on the context in which it is used Always be sure of the context in which a symbol is used before interpreting it 3 Some symbols such as x and 2 are of different types A value x of a variable X stands for an object in this case a number But 2 stands for an operation on the objects 2 cannot stand alone it must have another symbol with it in order to have meaning as a mathematical statement The same is true of symbols such as and which give relationships between objects Recognizing the type of symbol being used makes correct interpretation easier The analogy of objects as nouns and operations and relations as verbs might be useful But analogies shouldn t be pushed very far in mathematics 4 Remember that the language of mathematics relative to arithmetic operations on numbers has a grammar a set of rules that govern the manipulation of the language For instance when an expression involves both addition and multiplication the latter is done first as exhibited in the description above of what to do with Z x And mathematical language has punctuation the presence of parentheses that dictate in what order operations are to be done as used in the last expression given in Q26 Much of what is learned in algebra is just the grammar Students in this course are assumed to have thatknowledge but are encouraged to reinforce their understanding by asking appropriate questions when necessary 5 In constructing language it is useful to create connections between symbols That is the reason for the subscript x on the The subscript is there to connect the f to the x value with which it is paired a necessary connection since different xvalues can have different frequencies Another use of subscripts might arise when the values of a variable are restricted to those that are connected to subjects that fall in a particular category the subscript used designating the category 6 It is too often the case that part of the symbolic language is omitted For example Zxx x is often abbreviated to 2 x giving no reference to the variable X that has X s as its values But in the process the formula appears easier the xx perhaps an unnecessary complicating component clear from the context in which the formula is to be used WRITING ASSIGNMENT 2 Prepare a proposal for a statistical study Consulting the descriptions attached select one of the Course Data Sets In your proposal answer the following questions providing brief explanations for your anS us when appropriate including speculation when there is insufficient description hat a e the subjects under consideration at is the population under consideration hich of the data given is quantitative tGive symbolic names for variables for each set of quantitative data ie begin to create the mathematical language hat re all the potential values of eac variable viewed as population variables hich of the data is categorical What are the potential values of the categorical variables Make a list of questions for which answers can be investigated using the data including questions that involve combinations of variables For each question specify using the symbolic language created above what data will be useful Due on Mondav August 31 The IDEA of a DISTRIBUTION Frequency and Relative Frequency Quantitative data are numbers The same number can occur more than once in a 7 collection of data Q21 For the examples of Q3 what numbers are more likely to occur more often For a sample or a finite population the number of times a datum occurs can be counted the total is called the frequency for that particular datum A frequency distribution of quantitative data contains each numerical value in the data set paired with the frequency of that value Adding all of the frequencies for a data set gives how many pieces of data are in the set for samples the sample size The relative frequency of a number in a sample is its frequency divided by the sample size And the numerical values in a data set paired9 with their relative frequencies makes up a relative frequency distribution10 Q22 What advantage do relative frequency distributions have over frequency distributions The ideas of frequency and relative frequency distributions can be applied to categorical data as well Q23 How What is different Q24z Are there circumstances in which frequencies couldshould be considered data Variables and language To create a mathematical language to accompany these ideas define the following symbolic representations for a sample of quantitative data It is the sample size N is the population size if the population is finite X varies over all the numerical values in the sample or population for any value11 x of X fX is the frequency of that value The designation X is that of a variable and carries with it some ambiguity A variable X has both potential values and actual values The potential values are all those numbers that could conceivably occur in a sample that is the numbers that make up the population The actual values are the numbers that have been obtained collected those that occur in a sample That is X varies over the numbers under consideration and it is important to distinguish between variables for populations and samples Even for samples it is sometimes useful to restrict the values of a quantitative variable to the part of the sample that falls in a category giving an interplay between quantitative and categorical data In distributions a variable X can take on some or all values more than once So care has to be taken about whether X includes repetitions of its values or not Q26 Do categorical variables make sense Why or why not To create additional mathematical language let 2 mean add or take the sum of This symbol together with those defined above can be combined to create instructions for manipulating data For example Zxx K says add up all the values X of a variable X presumably allowing for repetitions and 2xx fxX says multiply each value of X by its frequency and then add all of the products presumably with no repeated values of X The subscript xx on 2 gives instructions on what values to add 9 For those comfortable with the mathematical idea of a function a distribution is a function with first coordinates data values and second coordinates frequencies of those data values 10Remember that it is not the individual pieces of data but rather the overall profile of all the data that is important in statistical analysis The focus above on individual pieces and their frequencies is just a step toward describing the overall profile that is the distribution Relative frequency distributions are also called probability distributions another reason for their importance 11 Note the distinction between lower and upper case Both are x s making a connection but the capital X stands for a distribution while the lowercase x is a value in the distribution a subtle but important distinction Q25 What would X x mean 2 Applicable in general to any finite collection of numbers