Descriptive+Statistics+for+Quantitative+Data+-+I-1

Descriptive+Statistics+for+Quantitative+Data+-+I-1 -...

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Unformatted text preview: Descriptive Statistics for Quantitative Data - I Notation • Suppose we have n experimental units and are interested in a single variables that is quantitative in nature. – Let xi denote the value of the variable for the i th experimental unit. • X = { x 1, x 2, . . . , xn } would be the collection of the observed values for the variable of interest for all n observations. – In other words, X is the dataset. Notation • Example 1: Suppose our dataset was A = {187.6, 125.3, 210.8, 157.3, 140.6}. Thus n = 5. • Could have x 1 = 187.6, x 2 = 125.3, x 3 = 210.8 x 4 = 157.3 and x 5 = 140.6. • Or could label x 1 = 140.6, x 2 = 125.3, x 3 = 157.3 Notation • In this instance (and throughout the course)the observation that we call x 1, the one we call x2 , and so on, is arbitrary. – It is merely just a convention to refer back to the observation. • In general it is not always arbitrary. An example would be if the observations occur in a sequence. – For instance, if the observations are recorded over time, the subscript would give Measures of Center Introduction • Question: Given the data { x 1, x 2, . . . , xn }, what would be a good guess as to what the value of a ‘typical’ observation is? • Answer: Measures of center provide an answer to the above question. • Three commonly reported measures of center are: the sample mode, sample median, and sample mean (sample Sample Mode • The sample mode is simply the observation that occurs the most frequently in the data set. • Example 2: Suppose the dataset is B = {2, 4, 6, 3, 2}. For this example the sample mode would be 2, since it occurs the most. • Problem – The sample mode may not be Sample Mode • Back to A = {187.6, 125.3, 210.8, 157.3, 140.6}. Here each observation occurs once and only once. • Example 3: Suppose the dataset is C = {2, 4, 4, 2, 6}. Here both 2 and 4 occur twice. • In both instances the sample mode is not unique. This happens more often than Sample Median • Intuitively, the sample median is the number that divides the data into two equal halves. However, this is not precise. A better definition is needed for the sample median. To do this, we will need to introduce another concept and notation, Order Statistics. Order Statistics • Given the data { x 1, x 2, . . . , xn }, define x ( i ) to be the ith smallest value. • So x (1) is the smallest value, x (2) the second smallest value, . . . , and x ( n ) the last smallest (i.e. the largest) value. • If two or more observations are tied, than necessarily the same number of order statistics must have the same value. Order Statistics • Back to Example 1: The data was x 1 = 187.6, x 2 = 125.3, x 3 = 210.8, x 4 = 157.3 and x 5 = 140.6....
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