Numbers & Data Basic Concepts

7/9/2013 Numbers & Data 2 2 Data Uses of Numbers Counting Finding Differences and Totals Averages and Percentages Rates Statistics and Probabilities

7/9/2013 Numbers & Data 3 3 Data Uses of Numbers Counting Miles driven Students in class Points on a test problem Finding totals Total minutes of phone usage this month Total semester hours this semester Total points scored this semester

7/9/2013 Numbers & Data 4 4 Data Uses of Numbers Finding differences Change from payment for lunch Price increase of gasoline Change in stock price Averages Test average for the semester Average price of gasoline in the U.S. Batting “average” for a baseball player

7/9/2013 Numbers & Data 5 5 Data Uses of numbers Percentages Percentage of students passing math Percent blood-alcohol by volume Percent increase in price Rates Rate of increase of water depth (feet/hour) Rate of miles driven per hour (miles/hour) Annual crime rate (crimes/year)

7/9/2013 Numbers & Data 6 6 Data Statistics Mean Same as average or first moment Median Central value Mode Most prevalent value Standard Deviation Average difference from mean

7/9/2013 Numbers & Data 7 Finding Averages Add n values, divide by n Call Avg the average of n values of x 7 Calculations with Data Avg = x 1 + x 2 + x 3 + ··· + x n n = n n 1  k = 1k = 1 xkxk

7/9/2013 Numbers & Data 8 Example Average of test scores 73, 85, 14, 92 Works fine for small number of values 8 Calculations with Data  k = 1k = 1 xkxk = 66 = ( ) 1 4 = (264) 1 4

7/9/2013 Numbers & Data 9 9 Example Travel d miles in t hours For d = 200 miles and t = 3 hours Works for large/infinite number of values Calculations with Data Average speed = d t = 200 miles 3 hours = mph

7/9/2013 Numbers & Data 10 Calculations with Data Finding percentages A per-100 proportion a as a percentage of number b is a is to b as P is to 100 a b (100) = P … that is a b = 100 P

7/9/2013 Numbers & Data 11 Calculations with Data Example In a chain saw 12 ounces of oil are added per gallon of gasoline What percentage of the mixture is oil ? Note that 1 gal = 128 oz For every 100 ounces of mixture ounces are oil % oil ≈ % = ( 100 ) %

7/9/2013 Numbers & Data 12 Calculations with Data Finding percent change A variable p changes by amount ∆p What percentage of p is ∆p ? Percent change in p is where p is the initial value ∆p p (100) %

7/9/2013 Numbers & Data 13 Calculations with Data Example The price of gasoline increased from $2.56 per gallon to $3.89 per gallon Change is: ∆p = 3.89 – 2.56 = 1.33 Percent change is ∆p p (100) = (100) ≈ %

7/9/2013 Numbers & Data 14 Calculations with Data Absolute value The unsigned “size” of a number Definition:  a = a, for a ≥ 0 – a, for a < 0 For any real number a the absolute value of a, written, is  a

7/9/2013 Numbers & Data 15 Calculations with Data Examples  7 = 7 = 3  0 = 0 Question: If a is any real number … … is – a positive or negative ?  – 3 so a = ( 3) = 3 – – – Here a = 3 –  – x 5 – – + =, for x ≥ 5, for x < 5

7/9/2013 Numbers & Data 16 Scientific Notation Standard notation – – Scientific notation x – x x 10 – 4 – x 10 – 1

7/9/2013 Numbers & Data 17 General form For real number r, scientific notation is r = c x 10 n where n is an integer and 1 ≤  Constant c usually a single-digit integer 17 Scientific Notation c < 10

7/9/2013 Numbers & Data 18 Think about it !