Yesterday s lab. Mean, Median, Mode. Section 2.3. Measures of Central Tendency. What are the units?

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1 Yesterday s lab Section 2.3 What are the units? Measures of Central Tendency Measures of Central Tendency Mean, Median, Mode Mean: The sum of all data values divided by the number of values For a population: For a sample: Median: The point at which an equal number of values fall above and fall below. If an even number then the average of the two middle values. Mode: The value with the highest frequency. If no repeated numbers, then no mode. 1

2 Additional Definitions Outlier: a data value that is far removed from other values (this will be quantified when we discuss data variation). Weighted Mean: mean of a data set whose values are not weighted equally. x w x w where w is the weight of each x value Example: An instructor recorded the average number of absences in a class during one quarter. For a random sample the data are: Calculate the mean, median, and mode Mean: Median: Calculating Metrics Sort data in order The middle value is 3, so the median is 3. Mode: The mode is 2 since it occurs the most times. Revised Metrics Suppose the student with 40 absences dropped the course. Calculate the mean, median and mode of the remaining values. Compare the effect of the change to each metric Calculate the mean, the median, and the mode. Mean: Shapes of Distributions Symmetric Uniform Mean = Median Median: Sort data in order The middle values are 2 and 3, so the median is 2.5. Mode: The mode is 2 since it occurs the most times. Skewed right Mean > Median Skewed left Mean < Median 2

3 Summary Definitions Other Distributions Symmetric data halves are mirror images Uniform all entries have equal frequencies (symmetric?) Skewed tail is elongated on one side Skewed left left tail elongated If skewed right, where does mean shift (relative to a symmetric distribution)? Section 2.4 Measures of Variation Definitions Range difference between minimum and maximum values Deviation difference between a value and the mean of a data set Population Variance mean of the squares of the deviations Population Standard Deviation square root of the variance 3

4 Two Data Sets Example: Closing prices for 2 stocks on 10 successive Fridays. Measures of Variation Range = Maximum value Minimum value Stock A Mean = 61.5 Median = 62 Mode = Stock B Mean = 61.5 Median = 62 Mode = Range for A = 56 = $11 Range for B = = $57 The range is easy to compute but only uses two numbers from a data set. Measures of Variation The deviation for each value x is the difference between the value of x and the mean of the data set. In a population, the deviation for each value x is: In a sample, the deviation for each value x is: Deviations Stock A Deviation The sum of the deviations is always zero. 4

5 Population Variance Population Variance: The sum of the squares of the deviations, divided by N. x deviation deviation Units? Sum of squares Population Standard Deviation Population Standard Deviation: The square root of the population variance. Units? The population standard deviation is $4.34. Sample Variance and Standard Deviation To calculate a sample variance divide the sum of squares by n 1. Summary Range = Maximum value Minimum value Population Variance The sample standard deviation, s, is found by taking the square root of the sample variance. Population Standard Deviation Sample Variance Sample Standard Deviation 5

6 Break and Quiz Question Break and Quiz Question mean = 1.244; median = 1.25; mode = 1.25 Sample standard deviation = Population standard deviation = Empirical Rule ( %) Data with symmetric bell-shaped distribution have the following characteristics. Using the Empirical Rule Example: The mean value of homes on a street is $125,000 with a standard deviation of $5,000. The data set has a bell shaped distribution. Estimate the percent of homes between $120,000 and $135, % 2.35% 13.5% 2.35% About 68% of the data lies within 1 standard deviation of the mean About 95% of the data lies within 2 standard deviations of the mean About 99.7% of the data lies within 3 standard deviations of the mean $120 thousand is 1 standard deviation below the mean and $135 thousand is 2 standard 68% % = 81.5% deviations above the mean. So, 81.5% have a value between $120 and $135 thousand. 6

7 Chebychev s Theorem For any distribution regardless of shape the portion of data lying within k standard deviations (k > 1) of the mean is at least 1 1/k 2. Chebychev s Theorem Example: The mean time in a women s 400 m dash is 52.4 s with a standard deviation of 2.2 s. Apply Chebychev s theorem for k = 2. Mark a number line in standard deviation units ( ) For k = 2, at least 1 1/4 = 3/4 or 75% of the data lie within 2 standard deviation of the mean. For k = 3, at least 1 1/9 = 8/9 = 88.9% of the data lie within 3 standard deviation of the mean. 2 standard deviations At least 75% of the women s 400-meter dash times will fall between 48 and 56.8 seconds. How much within 3 st devs? 1-1/9=8/9 (45.8 and 59 s) A Standard Deviation of Grouped Data Formula for the standard deviation for a frequency distribution: s x x n 1 2 f where n = f (number of entries in data set) Measures of Position Section 2.5 Multiply by the frequencies f See example 9 in text pg. 88 7

8 Quartiles Quartiles: divide the data into 4 equal parts. Q 1 = median of the data below Q2. Q 2 = median. Q 3 = median of the data above Q 2. Example: You are managing a store. The average sale for each of 27 randomly selected days in the last year is given. Find Q 1, Q 2, and Q Finding Quartiles The data in ranked order (n = 27) are: Median rank (27 + 1)/2 = 14. The median = Q2 = 42. There are 13 values below the median. Q1 rank = 7. Q1 is 30. Q3 is rank 7 counting from the last value. Q3 is 45. Interquartile Range difference between 3 rd and 1 st quartiles Q 3 Q 1 = = 15. (middle 50% of the data) Box and Whisker Plot Box and Whisker plot: 5 values to describe a data set. Q 1, Q 2, Q 3, minimum value, and maximum value Q 1 Q 2 = the median Q 3 Minimum value Maximum value Percentiles Percentiles - divide the data into 100 parts. There are 99 percentiles: P 1, P 2, P 3 P 99. P 50 = Q 2 = the median P 25 = Q 1 P 75 = Q 3 Interpretation: A 63rd percentile score means 63% of the scores and 37% of the scores (SAT or GRE scores) Interquartile Range = = 15 8

9 Percentiles Standard Scores Frequency (#) Standard Score or Z-Score - the number of standard deviations that a data value, x, falls from the mean. Growing Degree Day ( o C) Cumulative distributions can be used to find percentiles falls on or above 25 of the 30 values. 25/30 = So you can approximate 114 = P 83. The test scores for a civil service exam have a mean of 152 and standard deviation of 7. Find the standard z- score for a person with a score of: (a) 161 (b) 148 (c) 152 Calculations of z-scores (a) (b) (c) A value of x = 161 is 1.29 standard deviations above the mean. A value of x = 148 is 0.57 standard deviations below the mean. A value of x = 152 is equal to the mean. 9