Lecture 5 Dustin Lueker

2 Mode - Most frequent value. Notation: Subscripted variables n = # of units in the sample N = # of units in the population x = Variable to be measured x i = Measurement of the i th unit Mean - Arithmetic Average Median - Midpoint of the observations when they are arranged in increasing order STA 291 Summer 2010 Lecture 5

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 Sample ◦ Variance ◦ Standard Deviation  Population ◦ Variance ◦ Standard Deviation 4STA 291 Summer 2010 Lecture 5

5 1. Calculate the mean 2. For each observation, calculate the deviation 3. For each observation, calculate the squared deviation 4. Add up all the squared deviations 5. Divide the result by (n-1) Or N if you are finding the population variance (To get the standard deviation, take the square root of the result) STA 291 Summer 2010 Lecture 5

 If the data is approximately symmetric and bell-shaped then ◦ About 68% of the observations are within one standard deviation from the mean ◦ About 95% of the observations are within two standard deviations from the mean ◦ About 99.7% of the observations are within three standard deviations from the mean 6STA 291 Summer 2010 Lecture 5

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 The p th percentile (X p ) is a number such that p% of the observations take values below it, and (100-p)% take values above it ◦ 50 th percentile = median ◦ 25 th percentile = lower quartile ◦ 75 th percentile = upper quartile  The index of X p ◦ (n+1)p/100 8STA 291 Summer 2010 Lecture 5

 25 th percentile ◦ lower quartile ◦ Q1 ◦ (approximately) median of the observations below the median  75 th percentile ◦ upper quartile ◦ Q3 ◦ (approximately) median of the observations above the median 9STA 291 Summer 2010 Lecture 5

 Find the 25 th percentile of this data set ◦ {3, 7, 12, 13, 15, 19, 24} 10STA 291 Summer 2010 Lecture 5

 Use when the index is not a whole number  Want to start with the closest index lower than the number found then go the distance of the decimal towards the next number  If the index is found to be 5.4 you want to go to the 5 th value then add.4 of the value between the 5 th value and 6 th value ◦ In essence we are going to the 5.4 th value STA 291 Summer 2010 Lecture 511

 Find the 40 th percentile of the same data set ◦ {3, 7, 12, 13, 15, 19, 24}  Must use interpolation 12STA 291 Summer 2010 Lecture 5

 Five Number Summary ◦ Minimum ◦ Lower Quartile ◦ Median ◦ Upper Quartile ◦ Maximum  Example ◦ minimum=4 ◦ Q1=256 ◦ median=530 ◦ Q3=1105 ◦ maximum=320,000.  What does this suggest about the shape of the distribution? 13STA 291 Summer 2010 Lecture 5

 The Interquartile Range (IQR) is the difference between upper and lower quartile ◦ IQR = Q3 – Q1 ◦ IQR = Range of values that contains the middle 50% of the data ◦ IQR increases as variability increases  Murder Rate Data ◦ Q1= 3.9 ◦ Q3 = 10.3 ◦ IQR = 14STA 291 Summer 2010 Lecture 5

 Displays the five number summary (and more) graphical  Consists of a box that contains the central 50% of the distribution (from lower quartile to upper quartile)  A line within the box that marks the median,  And whiskers that extend to the maximum and minimum values  This is assuming there are no outliers in the data set 15STA 291 Summer 2010 Lecture 5

 An observation is an outlier if it falls ◦ more than 1.5 IQR above the upper quartile or ◦ more than 1.5 IQR below the lower quartile 16STA 291 Summer 2010 Lecture 5

 Whiskers only extend to the most extreme observations within 1.5 IQR beyond the quartiles  If an observation is an outlier, it is marked by an x, +, or some other identifier 17STA 291 Summer 2010 Lecture 5

 Values  Min = 148  Q1 = 158  Median = Q2 = 162  Q3 = 182  Max = 204  Create a box plot 18STA 291 Summer 2010 Lecture 5

 On right-skewed distributions, minimum, Q1, and median will be “bunched up”, while Q3 and the maximum will be farther away.  For left-skewed distributions, the “mirror” is true: the maximum, Q3, and the median will be relatively close compared to the corresponding distances to Q1 and the minimum.  Symmetric distributions? STA 291 Summer 2010 Lecture 519

 Value that occurs most frequently ◦ Does not need to be near the center of the distribution  Not really a measure of central tendency ◦ Can be used for all types of data (nominal, ordinal, interval)  Special Cases ◦ Data Set  {2, 2, 4, 5, 5, 6, 10, 11}  Mode = ◦ Data Set  {2, 6, 7, 10, 13}  Mode = 20STA 291 Summer 2010 Lecture 5

 Mean ◦ Interval data with an approximately symmetric distribution  Median ◦ Interval or ordinal data  Mode ◦ All types of data 21STA 291 Summer 2010 Lecture 5

 Mean is sensitive to outliers ◦ Median and mode are not  Why?  In general, the median is more appropriate for skewed data than the mean ◦ Why?  In some situations, the median may be too insensitive to changes in the data  The mode may not be unique 22STA 291 Summer 2010 Lecture 5

 “How often do you read the newspaper?” 23 ResponseFrequency every day969 a few times a week 452 once a week261 less than once a week 196 Never76 TOTAL1954 Identify the mode Identify the median response STA 291 Summer 2010 Lecture 5

 Statistics that describe variability ◦ Two distributions may have the same mean and/or median but different variability  Mean and Median only describe a typical value, but not the spread of the data ◦ Range ◦ Variance ◦ Standard Deviation ◦ Interquartile Range  All of these can be computed for the sample or population 24STA 291 Summer 2010 Lecture 5

 Difference between the largest and smallest observation ◦ Very much affected by outliers  A misrecorded observation may lead to an outlier, and affect the range  The range does not always reveal different variation about the mean 25STA 291 Summer 2010 Lecture 5

 Sample 1 ◦ Smallest Observation: 112 ◦ Largest Observation: 797 ◦ Range =  Sample 2 ◦ Smallest Observation: ◦ Largest Observation: ◦ Range = 26STA 291 Summer 2010 Lecture 5