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MATH 1342 CHAPTER 3 OH

MATH 1342 CHAPTER 3 OH - NUMERICALLY NUMERICALLY...

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Unformatted text preview: NUMERICALLY NUMERICALLY SUMMARIZING DATA NOTATION NOTATION N = SIZE OF POPULATION n = SIZE OF SAMPLE µ = MEAN OF POPULATION = MEAN OF SAMPLE X Σ = SUM OF INDIVIDUALS σ = POPULATION STANDARD DEVIATION S = SAMPLE STANDARD DEVIATION MEASURES OF MEASURES OF CENTRAL TENDENCY MEAN (or average) of a POPULATION : MEAN (or average) of a SAMPLE: N µ= ∑X i= 1 i N n X= ∑X i= 1 n i MEASURES OF MEASURES OF CENTRAL TENDENCY 5.3 5.5 5.6 5.7 5.7 5.8 5.9 6.2 6.3 6.3 6.4 6.6 6.6 6.7 6.8 7.1 7.1 7.3 7.6 7.9 n X= ∑X i =1 n i 128.4 = = 6.42 20 MEASURES OF MEASURES OF CENTRAL TENDENCY LAW OF LARGE NUMBERS: n→N AS THEN X →µ In other words, as the sample size gets closer to the population size the sample mean gets closer to the real population mean. MEASURES OF MEASURES OF CENTRAL TENDENCY MEDIAN: Middle value (if n is odd) or the average of the two middle values (if n is even). 5.3 5.5 5.6 5.7 5.7 5.8 5.9 6.2 6.3 6.3 6.4 6.6 6.6 6.7 6.8 7.1 7.1 7.3 7.6 7.9 MEDIAN=Average of 6.3 & 6.4 = 6.35 MEASURES OF MEASURES OF CENTRAL TENDENCY MODE: The most frequent value(s). Could be none or several. 5.3 5.5 5.6 5.7 5.7 5.8 5.9 6.2 6.3 6.3 6.4 6.6 6.6 6.7 6.8 MODES: 5.7, 6.3, 6.6, 7.1 7.1 7.1 7.3 7.6 7.9 MEASURES OF MEASURES OF CENTRAL TENDENCY SKEWED RIGHT FRE QUE NCY 20 15 10 5 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 VALUES MODE = 0.2 MEDIAN = 0.3 MEAN = 0.39 MEASURES OF DISPERSION MEASURES OF DISPERSION RANGE = MAXIMUM – MINIMUM 5.3 5.5 5.6 5.7 5.7 5.8 5.9 6.2 6.3 6.3 6.4 6.6 6.6 6.7 6.8 7.1 7.1 7.3 7.6 7.9 RANGE = 7.9 – 5.3 = 2.6 MEASURES OF DISPERSION MEASURES OF DISPERSION STANDARD DEVIATION can be thought of as the average distance of the values from the mean. N 2 Population: σ = ∑ i =1 ( X i − µ ) N Sample: ∑ (X n s= i =1 i −X n −1 ) 2 MEASURES OF DISPERSION MEASURES OF DISPERSION X X - MEAN = (X - MEAN)^2 5.3 5.3 - 6.42 = -1.12 1.25 5.5 5.5 - 6.42 = -0.92 0.85 * * * * * * * * * * * * 7.3 7.3 - 6.42 = 0.88 0.77 7.6 7.6 - 6.42 = 1.18 1.39 7.9 7.9 - 6.42 = 1.48 2.19 SUM(X-MEAN)^2 /(n-1) = 0.53 SQRT[SUM(X-MEAN)^2 /(n-1)] = 0.73 MEASURES OF DISPERSION MEASURES OF DISPERSION ALTERNATIVE FORMULAS: 2 σ= N Xi ÷ ∑ N 2 ∑X i − i =1 N i= 1 N 2 Xi ÷ 2 ∑ n ( 128.4 ) X i2 − i =1 834.44 − ∑ n 20 s = i =1 = n −1 19 n Variation = σ 2 MEASURES OF DISPERSION MEASURES OF DISPERSION VARIANCE: The square of the Standard Deviation. Population VARIANCE = σ Sample VARIANCE = s 2 2 SYMBOL SUMMARY SYMBOL SUMMARY ITEM POPULATION PARAMETER SAMPLE STATISTIC N n s MEAN µ x STANDARD DEVIATION σ s VARIANCE σ2 s2 SIZE USING THE CALCULATOR USING THE CALCULATOR STAT EDIT: Enter Data Into L1 STAT CALC 1: 1­Var Stats ENTER 2nd “1” (L1) ENTER EXAMPLE EMPERICAL RULE EMPERICAL RULE Approximately 68% of the population are within the range of ± 1 Approximately 95% of the population are ±2 within the range of Approximately 99.7% of the population are within the range of ± 3 µ µ σ µ OVERHEAD σ σ CHEBYSHEV’S INEQUALITY CHEBYSHEV’S INEQUALITY THE PERCENT OF THE POPULATION WITHIN +/­ K STANDARD DEVIATIONS OF THE MEAN IS GIVEN BY 1 1 − 2 ÷*100% K EXAMPLE: IF K=2.5 THEN 1 1 − ÷*100% = ( 1 − 0.4 ) *100% = 60% 2.5 OF THE POPULATION IS WITHIN +/­ 2.5 STANDARD DEVIATIONS OF THE MEAN MEAN AND STANDARD MEAN AND STANDARD DEVIATION FROM FREQUENCY DISTRIBUTION MEAN: µ = ∑( x * f ) (X *f )÷ ∑ f ∑ ( X − µ ) * f ∑ ( X * f ) − ∑ i i N N i 2 N STANDARD DEVIATION: = σ 2 i =1 2 i ∑f i i = i =1 2 i i =1 i ∑f i ∑f i IF FROM CONTINUOUS FREQUENCY DISTRIBUTION, USE THE MIDPOINT FROM EACH CLASS. TO DO ON CALCULATOR, ENTER TABLE IN L1 & L2. THEN DO STAT CALC 1: 1­Var Stats ENTER L1,L2 i i WEIGHTED AVERAGE WEIGHTED AVERAGE Calculated like mean for frequency distribution. ∑( X * f ) µ= ∑f i i i Example using Grade Point Average GRADE GRADE VALUE (x) CREDIT HOURS (f) x*f C 2 3 6 B 3 4 12 A 4 3 12 A 4 2 8 B 3 4 12 16 50 50 µ = = 3.1 16 MEASURES OF RELATIVE MEASURES OF RELATIVE POSITION Z­SCORE: X −µ X−X Z= = σ s MEASURES OF RELATIVE MEASURES OF RELATIVE POSITION Used to compare relative position of data in two separate groups. “A” has a score of 78 in a class with a mean of 84 and a std. dev. of 6. “B” has a score of 86 in a class with a mean of 90 and a std. dev of 3. Who did better relative to their class? MEASURES OF RELATIVE MEASURES OF RELATIVE POSITION Percentile: The value for which k% of the data set is ≤ Pk. For instance if P18=7.6, then 18% of the sample or population is less than or equal to 7.6 If your MATH SAT score was in the 92 percentile, then 92% of the population had a score less than OR equal to yours. MEASURES OF RELATIVE MEASURES OF RELATIVE POSITION Three important percentiles: P25 = Q1: 25% of the data ≤ Q1 P50 = Q2: 50% of the data ≤ Q2 (median) P75 = Q3: 75% of the data ≤ Q3 MEASURES OF RELATIVE MEASURES OF RELATIVE POSITION FIVE NUMBER SUMMARY MIN, Q1, MEDIAN, Q3, MAX BOX PLOT MEASURES OF RELATIVE MEASURES OF RELATIVE POSITION 5.3 5.5 5.6 5.7 5.7 5.8 5.9 6.2 6.3 6.3 Using Calculator: 6.4 6.6 6.6 6.7 6.8 7.1 7.1 7.3 7.6 7.9 STAT EDIT: Enter Data Into L1 STAT CALC 1: 1­Var Stats ENTER 2nd “1” (L1) ENTER UNUSUAL VALUES UNUSUAL VALUES Inter Quartile Range: IQR = Q3 – Q1. µ− IQR 1.5 Any Value Less Than Is Considered Unusual (called lower fence). µ +1.5 IQR Any Value Greater Than Is Considered Unusual (called upper fence). UNUSUAL VALUES UNUSUAL VALUES ALSO CONSIDER A VALUE TO BE UNUSUAL IF ITS Z­SCORE IS LESS THAN – 2. OR X­SCORE IS GREATER THAN +2. ...
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