Unit 2: Numerical Descriptive Measures

Transcription

1 Unit 2: Numerical Descriptive Measures Summation Notation Measures of Central Tendency Measures of Dispersion Chebyshev's Rule Empirical Rule Measures of Relative Standing Box Plots z scores Jan 28 10:48 AM Def 1: Numerical Descriptive Measures describe numerical data in terms of summary properties We will study 1. Measures of Central Tendency 2. Measures of dispersion 3. Measures of relative position Before we get started we need to remember Summation Notation Example 1: Suppose a data set has n = 5 observations: {5, 6, 2, 4,7} find So what would I do if I wanted Jan 28 10:59 AM

2 Example 2: Example 3: Jan 31 10:46 AM Jan 31 10:51 AM

3 1. Measures of Central Tendency Def 2: Measures of Central Tendency describe the center or typical value of the data. Typical observations in data sets include Mode, Mean, and Median Def 3: Mode the measure that occurs most often. Type 1: Type 2: Type 3: Jan 31 10:52 AM Def 4: Arithmetic Mean, or Mean is the average of the data values i.e. sum of the observations divided by the sample size (n) ( x bar is what we use to represent a sample mean) Example 4: Find the mean of the following set of data n = Jan 31 10:59 AM

4 Example 5: Let's take the same data set and change one of the data points to an outlier. Jan 31 11:20 AM Def 5: resistant measure a summary measure that is not affected by extreme observations. Def 6: median (m) the middle observation of a data set, after arranging the data in ascending order. Example 6: Find the median of the data set: 5, 6, 2, 4, 7 1. Arrange observations in ascending order 2, 4, 5, 6, 7 2. Find the number of observations if n is odd, the median is the middle observation if n is even, the median is the average of the two middle observations n = 5 (odd) median = 5 3. the location of the median is found using Jan 31 11:23 AM

5 Example 7: Find and compare the Mean, Median and Mode of the following. 55, 88, 99, 87, 110, 210, 65, 100, 75, arrange in ascending order 55, 55, 65, 75, 87, 88, 99, 100, 110, n = 10 (even) Mean = 94.4 Median = 87.5 Mode = 55 Note: mode<median<mean Jan 31 12:41 PM Types of Distributions 1. Symmetric 2. Skewed Right 3. Skewed Left Mean < Median<Mode Helpful pneumonic: Mean comes before median in the dictionary and will dictate skewness. if median<mean, skewed to the right if mean<median, skewed to the left. The median is less sensitive to extreme observations; it is therefore a more resistant measure than the mean. Jan 31 12:45 PM

6 (mu) (eta) Jan 31 1:00 PM Def 7: trimmed mean is the arithmetic mean of observations after you trim off a percentage α from each end of the ordered data Example 7: (continued) A. find the 10% trimmed mean of 55, 55, 65, 75, 87, 88, 99, 100, 110, compute (0.1)n = (0.1)(10) = 1 therefore omit one observation from each end 2. find the mean of the remaining 8 observations B. find the 15% trimmed mean Jan 31 12:53 PM

7 2. Measures of Dispersion Def 8: measures of dispersion measure to what extent data values are spread out about the center. Typical observations in the data set include range, variance, standard deviation and inter quartile range. Example 8: Compare the mean and median of the following sets of data. *write down data we will use it for multiple definitions Mean = Median = 45 These three data sets have the same center, but different spreads. Jan 31 1:06 PM Def 9: range the difference between the largest and smallest observations Example 8 (continued): Find the range for data sets 1, 2, and 3 The range is the same for sets 1 and 2. Remember the center is 45 for each data set note that the range is clearly no resistant Jan 31 1:14 PM

8 We will now define the standard deviation and variance of a data set consisting of observations x 1 x n Def 10: devation from sample mean The absolute value of the deviance tells how far an observation lies away from the mean. The sign indicates which direction, left ( ) or right (+) Jan 31 1:23 PM Def 11: sample standard deviation: the distance of a typical value from the mean Def 12: sample variance s 2 is the square of the sample standard deviation s. Example 8 (continued): calculate the standard deviation of the problem sets Data Set Value (x 45) (x 45)^2 Sum sum/ (n 1) sqrt(sum/ (n 1)) Jan 31 1:27 PM

9 Example 9: The amount of radiation received at a greenhouse plays an important role in determining the rate of photosynthesis. The accompanying observations on incoming solar radiation were read from a graph in the paper "Radiation Components over Bare and Planted Soils in a Greenhouse" Jan 31 1:59 PM Example 9: Answers Jan 31 2:12 PM

10 Chebychev's Rule and Emperical Rule As Statisticians we combine measures of central tendency with measures of variability to summarize a distribution for population or sample data sets. Using these rules, it is possible to interpret the standard deviation and decide what proportions of observations generally are within 1 standard deviation of the mean 2 standard deviations of the mean Jan 31 3:35 PM Chebyshev's Rule Knowing and s for a sample data set, Chebyshev's rule give information on the proportion of observations that fall within a specific number of standard deviations from the mean, i.e., the proportion of observations that are where k is a number greater than or equal to 1, i.e., k 1. Jan 31 3:42 PM

11 Rule: Let k be a number greater than or equal to 1. The proportion of observations within k standard deviations of the mean, i.e. within Chebyshev's rule does not depend on the shape of the distribution and applies to any data set. Jan 31 3:54 PM Example 10: Chebyshev's Rule for Starting Salaries (in thousands) data. What percentage of observations lie within 2.7 StDev of the mean? Jan 31 3:58 PM

12 11 Feb 1 12:53 PM Empirical Rule applies only when the frequency distribution is mound shaped or bell shaped. (for normal distributions) Rule: 1. Approximately 68% of the observations lie within 1 sd's 2. Approximately 95% of the observations lie within 2 sd's 3. Approximately 99.7% of the observations lie within 3 sd's Jan 31 4:06 PM

13 12 Jan 31 4:14 PM 13 Feb 1 1:02 PM

14 3. Measures of Relative Position Def 13: Measure of relative position/standing describe how a data value relates to other data values in a given data set. Typical observations in the data set include percentiles and quartiles Def 14: pth percentile a value such that p percent of the observations in the data set fall at or below that value. ex 95% of all test scores are at or below 650, whereas only 5% are above 650, then 650 is called the 95th percentile of the data set. Procedure to find the pth percentile Jan 31 4:19 PM Example 10 (continued): Starting Salaries (in thousands) data. Feb 1 12:33 PM

15 Quartiles Def 15: first (lower) quartile [Q 1 or Q L or P 25 ] 25th percentile i.e. 25% of the data is below it Def 16: middle quartile [m] 50th percentile or median i.e. 50% of the data is below it Def 17: upper quartile [Q 3 or Q U or P 75 ] 75th percentile i.e. 75% of the data is below it You can also obtain the quartiles by dividing the n ordered observations into a lower half and an upper half and find the median of each half *if n is odd, the median is excluded from both halved when computing quartiles Feb 1 12:40 PM Box Plots Def 18: interquartile range Q U Q L, it is a measure of variability that is not sensitive to the presence of outliers unlike the standard deviation. Def 19: When to use: To highlight the center, spread, or any outliers in the data How to Construct: 1. Draw a measurement scale 2. Construct a box with the ends (hinges) at QL and QU. Show the median (Q2) in the box. 3. From each hinge, calculate distance of 1.5 (IQR) 4. Whiskers are drawn from each hinge to most extreme observations inside the inner fence. 5. From each hinge, calculate distances 3.0(IQR) 6. If an observation in the data set falls between the inner and outer fences, it is a mile outlier 7. Those falling outside the outer fence, are extreme outlier Feb 1 1:07 PM

16 Example 10 (continued): Using the given data construct a box plot. smallest observation = 5.2 largest observation 13.5 Feb 1 1:39 PM Def 20: population z score for x is μ is the population mean σ is the population standard deviation Since population z scores are not known exactly we use Def 21: sample z score which gives the distance in standard deviations between the observation x and the mean. Feb 3 9:19 AM

17 Example 14: compare 2 job offers: because it is.5 standard deviations above the mean of 'all' marketing jobs Feb 3 9:23 AM Feb 3 9:24 AM

18 Example 15: A parking lot owner's receipts for 100 days had a mean of $360 and standard deviation of $25 Use z scores to coment about the difference in Yesterday's receipt $370 Today's receipt $460 Today's Feb 3 9:25 AM 6 Feb 3 9:33 AM