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# MATH 121 STAT Study Guide Test #1 MAT 121

WCUPA

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This 4 page Study Guide was uploaded by Cara Landis on Monday February 15, 2016. The Study Guide belongs to MAT 121 at West Chester University of Pennsylvania taught by Dr. Kump in Winter 2016. Since its upload, it has received 148 views. For similar materials see Intro to Statistics in Mathematics (M) at West Chester University of Pennsylvania.

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I was sick all last week and these notes were exactly what I needed to get caught up. Cheers!

-*Kaela*

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Date Created: 02/15/16

January 27, 2016 (Wed) MAT-121 Ch. 2 Quantitative & Qualitative Data Quantitative- “quantity”, numerical Qualitative- categorical answer Qualitative Data (2.1) Class- a category for qualitative data Class Frequency- number of observations in data set that fall into a certain class Class Relative Frequency- class frequency divided by total number of observations in data set Class Percentage- class relative frequency x100 Qualitative Data: Graphical Methods Pie Chart, Bar Graph, and Pareto Diagram Quantitative Data: Graphical Methods (2.2) Dot plots, stem-and-leaf diagram, histograms Dot Plots: horizontal scale, good for small data sets Stem and Leaf Diagrams: divides values, good for small data sets Histograms: larger data sets, but not precise, able to control amount of info displayed. Frequencies or Relative Frequencies are shown in a class interval Summation Notation (2.3) Summation symbol () tells us to add all the values of variables from the first (x) to the last (x) n X =x +x +x +...+x i=1 i 1 2 3 n Measures of Central Tendency (2.4) Central Tendency- numerical value around which data tends to cluster Variability- how strong the data cluster around the values- spread of the data Mean- average of a data set; take sum of x-values dividing by “n” n ´= ∑ Xi i=1 n Find Mean: 42000, 35000, 82000, 65000, 49000, 58000, 75000, 378000000 Mean: 47,300,750 Find Median: 42000, 35000, 82000, 65000, 49000, 58000, 75000, 378000000 Median: 61,500 Median is better to use with skewed data. Median- value located in the middle of a set of quantitative data Find Median: 5,2,7,1,4,3,2 Rearrange Data: 1,2,2,3,4,5,7 Median: 3 Skewed- mean is pulled above or below the median depending on the unusual value(s). Mode: value that occurs most in a set of data Find Mode: 6,7,2,3,4,6,2,6 Mode: 6 Modal Class: class with highest relative frequency (highest bar on bar graph) Numerical Measures of Variability (2.5) Mean, median, and mode give us an idea of the “middle” of a set of data. Variability gives us an idea of how spread out data is around middle. Range: equal to the largest value minus the smallest value. (not informative) Variance: average squared distance the data is from the average Sample 1: 1,2,3,4,5 Sample 2: 2,3,3,3,4 Mean: 3 Mean: 3 Variance: 2.5 Variance: .5 Sample Standard Deviation: equal to positive square root of sample variance. s= √ 2 . Interpret Standard Deviation (2.6) Chebyshev’s Rule: applies to ANY data set, provides lower bound for percentages, percentages will be much higher most of the time. 1 For any number k>1, at least ( ) 2 of observations will lie within k k standard deviations of the mean. Empirical Rule: bell shaped and symmetric- refine Chebyshev’s Theorem into more precise 68% of data falls within one standard deviation of mean 95% of data falls within 2 standard deviations of mean 99.7% of data falls within 3 standard deviations of mean Inferences using Empirical Rule: If an observation falls more than 2 standard deviations of the mean, we call it a rare event. o Only 5% of the time If we have an observation more than 3 standard deviations from the mean, it is very rare. o Only 0.3% of the time Standard Deviation and Range: s ≈range/4 (if we only know the range) Numerical Measures of Relative Standing (2.7) Measures of Relative Standing: descriptive measures of relationship of a measurement to rest of data Quartiles: percentiles that partition data into 4 categories Lower Quartile: 25 percentile Middle Quartile: 50 percentile th Upper Quartile: 75 percentile Z-score: how many standard deviations above/below the mean a particular measurement is, also allows us to remove original units on data & determine relative position in units of SD Z = x−´x s Ex) the beat of hummingbird wings in flight averages 55 times per second Assume the SD is 10 and distribution is symmetrical & mounded. An individual hummingbird is measured with 75 beats per second. x−´x What is the bird’s z-score? Z = s z= 75−55 = 2 10 Z-scores are related to the empirical rule: 68% between -1 and 1 95% between -2 and 2 99.7% between -3 and 3 Outlier: measurement that is unusually large or small relative to other values Three possible causes: Observation, recording or data Item is from a different population A rare, chance event Tools for Detecting Outliers: Box Plot relies on the interquartile rang (IQR), Q Q difference between upper and lower quartiles. IQR= 3 - 1 Detecting outliers using Box Plot 1. Fine Q L and QU 2. Calculate the IQR 3. Find the lower and upper inner fences a. Q1-1.5*IQR b. Q3+1.5*IQR 4. Find the lower and upper outer fences a. Q1-3*IQR b. Q3+3*IQR Inner Fences: Q 1 -1.5IQR= 35.65-1.5(2.7)= 31.6 Q 3 +1.5IQR=38.35+1.5(2.7)= 42.4 Upper Fences: Q 1-3IQR= 35.65-3(2.7)= 27.55 Q 3 +3IQR= 38.35+3(2.7)= 46.45

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