Applied Statistics MATH 170
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Date Created: 10/29/15
MATH 170 Applied Statistics Chapter 4 Numerical Methods for Describing Data Important Terms and Concepts Some initial notation o z the variable we have sample data for o n the sample size 0 m1 m2 7mn the n observations 0 Ex 1 2 an the sum of the observation values I Section 41 Describing the Center of a Data Set The focus two ways for describing the center of a numerical data set mean and median De nition 1 Mean The sample mean denoted i is the average of the sample values 12 n7amp 71 7 71 j The population mean is denoted by a and is the mean of the entire population De nition 2 Median The sample median is the middle value of the ordered sample values Ifn is odd then the median is the middle value but ifn is even the median is the average of the two middle values The population median is the median for the entire population De nition 3 Skewness If the histogram of the data is unimodal then 0 symmetric histograms indicate the mean median o positively skewed histograms indicate the mean gt median o negatively skewed histograms indicates the mean lt median De nition 4 Trimmed Mean A trimmed mean is a mean computed after deleting a selected number of values from each end of the ordered data list The trimming percentage is the percent of the n values deleted from each end of the ordered list before computing the mean II Section 42 Describing Variability in a Data Set The focus methods of showing the spread or variability in a data set deviations from the mean variance and standard deviation De nition 5 Deviations from the Mean The n deviations from the sample mean are the di erences m17i m27i zn7m If mk 7 i is positive then mk is greater than i and if mk 7 i is negative then mk is less than i De nition 6 Variance The sample variance denoted by s2 is given by 2 295 7 zt ziiyt i2 S n71 7 n71 ie the sum of the squared deviations from the mean divided by n 7 1 The population variance is denoted by 0392 De nition 7 Standard Deviation The sample standard deviation denoted by s is the square root of the variance ie s The standard deviation may be thought of as a measure of how far the observations are from the mean i on average The population standard deviation is denoted by a W De nition 8 Quartiles and Interquartile Range The lower quartile separates the bottom 25 of the data from the top 75 39 ie the median of the lower half of the sample 0 The upper quartile separates the top 25 of the data from the bottom 75 39 ie the median of the upper half of the sample The middle quartile is the median The interquartile range iqr upper quartile lower quartile Note ifn is odd the median is included in both halves III Section 43 Summarizing a Data Set Boxplots Boxplots are yet another way to quickly and easily visualize information about the data namely it s median quartile values and its range including identifying outliers De nition 9 Outliers 0 An observation is an outlier if it is more than 15iqr away from the nearest odd quartile in terms of a box plot this will be the end of one of the boxes 0 An outlier is extreme if it is more than 3iqr ie twice the outlier minimum away from the nearest odd quartile o If an outlier isn t extreme it is considered mild o The interquartile range iqr upper quartile lower quartile Note ifn is odd the median is included in both halues Constructing a Box Plot 1 First nd the median lower quartile upper quartile and calculate the interquartile range iqr 2 Above a number line draw a rectangular box with the left end at the lower quartile and the right end at the upper quartile Draw a vertical line at the median not the meanl 3 Place marks at distances 15iqr to the left of the left end and to the right of the right end of the box These are the inner fences Similarly place marks for the outer fences at distances 3iqr from each end 4 Extend horizontal lines whiskers from each end of the box out to the most extreme observa tions still within the inner fences 5 Mild outliers are observations between the inner and outer fences Show them as shaded circles Extreme outliers those observations beyond the outer fences are shown as open circles III Section 44 Interpreting Center and Variability The focus combining info on the center and variability of a data set Notion of being within k standard deviations of the mean Theorem 1 Chebyshev7s Rule 1 For k gt 1 the proportion of observation within k standard deviations is at least 1 7 Note this is a conservative estimate Theorem 2 Empirical Rule If the histogram of values in a data set can be reasonably well approximated by a normal curve then 0 roughly 68 of the observations are within 1 standard deviation of the mean 0 roughly 95 of the observations are within 2 standard deviations of the mean 0 roughly 997 of the observations are within 5 standard deviations of the mean Observations within k Standard Deviations of i for k value Chebyshev s Rule Empirical Rule 1 2 3 De nition 10 ZScore The zscore corresponding to a particular observation gives the number of standard deviations between the observation and the mean observation mean pk 7 i 2 score 7 standard deviation 3 If the 2 score is positive the observation is above the mean and if it is negative the observation is below the mean De nition 11 Percentiles IfO r g 100 the rth percentile is the value such that r percent of the observations in the data set fall at or below that value MATH 170 Applied Statistics Using the TI 83 for Descriptive Statistics with Numerical Data Putting the Data in Lists 1 Go to the STAT menu and select option number 17 Edit 2 Clear out any entries in your lists by going to the top of each column using the up arrow7 hitting CLEAR7 then ENTER 3 Put the data in the rst list L1 4 Once the data has been entered quit this screen by hitting 2nd QUIT Generating a Histogram of the Data in L1 1 Clear out any functions you have in the Y list7 or turn their graphs off 2 Go to your STAT PLDT menu 2nd Y 3 Select the rst plot7 Plotl and hit ENTER 4 Highlight the appropriate options by using your cursor to move among the choices and hit ENTER when your cursor is over the desired option You7ll want 0 On for the rst line 0 Select the bar chart icon for the Type 0 Put L1 in the Xlist row by typing 2nd 1 0 Leave the Freq as 1 5 Hit Zoom Stat7 ie ZUDM 97 to View your histogram 6 NOTE Go to the WINDOW menu The Xscl value determines the width of each bar7 beginning with Xmin You can adjust Xscl to be whatever value you desire7 then select GRAPH to return to your new histogram 7 ALSO NOTE When looking at the graph of the histogram7 you can hit TRACE to trace along the bars The class interval goes from the listed min value through the max value that appear on the lower left of your screen The frequency for each class is given by the 11 value at the lower right of the screen Generating a Scatter Plot of the Data in L1 and L2 Use the same instructions as you did for the Histogram7 but select the rst TYPE icon Doing this will also add a Ylist below the Xlist Put L1 in the Xlist and L2 in the Ylist NOTE After you are done with your statistical plotting7 it is best to turn your STAT PLDTS off This can mess up your function graphing later on7 and you may not realize it Calculating Mean Median etc for Data in L1 1 Go to the STAT menu and move right with your cursor to the CALC menu and select the rst option7 1 Var Stats This will take you back to the main calculator screen 2 Press ENTER 3 The calculator will show you a list of information to scroll through i is the mean of the data Ex is the sum of the data Ex2 is the sum of the squared data values Sm is the standard deviation value for the data sample in is the estimated standard deviation for the population IGNORE THIS ONE 71 is the number of items in the data set mmX is the minimum data value Q1 is the lower quartile value note7 the calculator does NOT include the median in guring the quartiles if you have an odd number of data points7 but the method in our text does Med is the median value Q3 is the upper quartile value see note for Q1 maxX is the maximum data value MATH 170 Applied Statistics Section 74 Mean and Standard Deviation of Random Variables Important Terms and Concepts 1 Mean And Standard Deviation De nitions De nition 1 Mean of a Random Variable The mean value of a random variable x denoted by um describes where the probability distribu tion ofz is centered Sometimes the term ewpected value is used as a synonym for the mean and is denoted by De nition 2 Standard Deviation of a Random Variable The standard deviation of a random variable x denoted by am describes the variability in the probability distribution ofz is centered 2 Calculating um and 0 for Discrete Random Variables De nition 3 Mean of a Discrete Random Variable The mean value of a discrete random variable x denoted by um is computed by multiplying every possible value ofz by it s associated probability px then adding those quantities up Hz 2 WW all m values De nition 4 Variance and Standard Deviation of a Discrete Random Variable The variance of a discrete random variable x denoted by 0 is computed by rst nding the variance between each individual m value and the mean x 7 u then multiplying each of those values by the associated probability Those values are then added up 05 Z 05 i W WM all m values The standard deviation of m denoted by am is the square root of the variance am V variance a Homework pp 378 380 28 31 33 34
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