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## Statistics with Economics and Business Applications

содержание презентации «Statistics with Economics and Business Applications.ppt»
 Сл Текст Сл Текст 1 Statistics with Economics and Business 21 estimate the population standard deviation Applications. Chapter 2 Describing Sets of s. Dividing by n –1 gives us a better Data Descriptive Statistics – Numerical estimate of s. Measures. 22 Measures of Relative Standing. How 2 Review. I. What’s in last lecture? many measurements lie below the Descriptive Statistics – tables and measurement of interest? This is measured graphs. Chapter 2. II. What's in this by the pth percentile. (100-p) %. p %. lecture? Descriptive Statistics – 23 Examples. 90% of all men (16 and Numerical Measures. Read Chapter 2. older) earn more than \$319 per week. ? 3 Describing Data with Numerical Median. ? Lower Quartile (Q1). ? Upper Measures. Graphical methods may not always Quartile (Q3). \$319 is the 10th be sufficient for describing data. percentile. BUREAU OF LABOR STATISTICS Numerical measures can be created for both 2002. populations and samples. A parameter is a 24 Quartiles and the IQR. The lower numerical descriptive measure calculated quartile (Q1) is the value of x which is for a population. A statistic is a larger than 25% and less than 75% of the numerical descriptive measure calculated ordered measurements. The upper quartile for a sample. (Q3) is the value of x which is larger 4 Measures of Center. A measure along than 75% and less than 25% of the ordered the horizontal axis of the data measurements. The range of the “middle distribution that locates the center of 50%” of the measurements is the the distribution. interquartile range, IQR = Q3 – Q1. 5 Some Notations. We can go a long way 25 Calculating Sample Quartiles. The with a little notation. Suppose we are lower and upper quartiles (Q1 and Q3), can making a series of n observations. Then we be calculated as follows: The position of write as the values we observe. Read as Q1 is. The position of Q3 is. once the “x-one, x-two, etc” Example: Suppose we measurements have been ordered. If the ask five people how many hours of they positions are not integers, find the spend on the internet in a week and get quartiles by interpolation. the following numbers: 2, 9, 11, 5, 6. 26 Example. Q1is 3/4 of the way between Then. the 4th and 5th ordered measurements, or 6 Arithmetic Mean or Average. The mean Q1 = 65 + .75(65 - 65) = 65. The prices of a set of measurements is the sum of the (\$) of 18 brands of walking shoes: 60 65 measurements divided by the total number 65 65 68 68 70 70 70 70 70 70 74 75 75 90 of measurements. where n = number of 95. Position of Q1 = .25(18 + 1) = 4.75 measurements. Position of Q3 = .75(18 + 1) = 14.25. 7 Example. Time spend on internet: 2, 9, 27 Example. Q3 is 1/4 of the way between 11, 5, 6. If we were able to enumerate the the 14th and 15th ordered measurements, or whole population, the population mean Q3 = 75 + .25(75 - 74) = 75.25. and IQR = would be called m (the Greek letter “mu”). Q3 – Q1 = 75.25 - 65 = 10.25. The prices 8 Median. The median of a set of (\$) of 18 brands of walking shoes: 60 65 measurements is the middle measurement 65 65 68 68 70 70 70 70 70 70 74 75 75 90 when the measurements are ranked from 95. Position of Q1 = .25(18 + 1) = 4.75 smallest to largest. The position of the Position of Q3 = .75(18 + 1) = 14.25. median is. once the measurements have been 28 Using Measures of Center and Spread: ordered. The Box Plot. Divides the data into 4 sets 9 Example. The set: 2, 4, 9, 8, 6, 5, 3 containing an equal number of n = 7 Sort: 2, 3, 4, 5, 6, 8, 9 Position: measurements. A quick summary of the data .5(n + 1) = .5(7 + 1) = 4th. The set: 2, distribution. Use to form a box plot to 4, 9, 8, 6, 5 n = 6 Sort: 2, 4, 5, 6, 8, 9 describe the shape of the distribution and Position: .5(n + 1) = .5(6 + 1) = 3.5th. to detect outliers. The Five-Number 10 Mode. The mode is the measurement Summary: Min Q1 Median Q3 Max. which occurs most frequently. The set: 2, 29 Constructing a Box Plot. The 4, 9, 8, 8, 5, 3 The mode is 8, which definition of the box plot here is occurs twice The set: 2, 2, 9, 8, 8, 5, 3 similar, but not exact the same as the one There are two modes—8 and 2 (bimodal) The in the book. It is simpler. Calculate Q1, set: 2, 4, 9, 8, 5, 3 There is no mode the median, Q3 and IQR. Draw a horizontal (each value is unique). line to represent the scale of 11 Example. The number of quarts of milk measurement. Draw a box using Q1, the purchased by 25 households: 0 0 1 1 1 1 1 median, Q3. 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5. Mean? 30 Constructing a Box Plot. *. Isolate Median? Mode? (Highest peak). outliers by calculating Lower fence: 12 Extreme Values. The mean is more Q1-1.5 IQR Upper fence: Q3+1.5 IQR easily affected by extremely large or Measurements beyond the upper or lower small values than the median. The median fence is are outliers and are marked (*). is often used as a measure of center when 31 Constructing a Box Plot. Draw the distribution is skewed. “whiskers” connecting the largest and 13 Extreme Values. Symmetric: Mean = smallest measurements that are NOT Median. Skewed right: Mean > Median. outliers to the box. Skewed left: Mean < Median. 32 Example. Amount of sodium in 8 brands 14 Measures of Variability. A measure of cheese: 260 290 300 320 330 340 340 along the horizontal axis of the data 520. distribution that describes the spread of 33 Example. IQR = 340-292.5 = 47.5 Lower the distribution from the center. fence = 292.5-1.5(47.5) = 221.25 Upper 15 The Range. The range, R, of a set of n fence = 340 + 1.5(47.5) = 411.25. Outlier: measurements is the difference between the x = 520. largest and smallest measurements. 34 Interpreting Box Plots. Median line in Example: A botanist records the number of center of box and whiskers of equal petals on 5 flowers: 5, 12, 6, 8, 14 The length—symmetric distribution Median line range is. R = 14 – 5 = 9. Quick and easy, left of center and long right but only uses 2 of the 5 measurements. whisker—skewed right Median line right of 16 The Variance. The variance is measure center and long left whisker—skewed left. of variability that uses all the 35 Key Concepts. I. Measures of Center 1. measurements. It measures the average Arithmetic mean (mean) or average a. deviation of the measurements about their Population mean: m b. Sample mean of size mean. Flower petals: 5, 12, 6, 8, 14. n: 2. Median: position of the median = 17 The Variance. The variance of a .5(n +1) 3. Mode 4. The median may be population of N measurements is the preferred to the mean if the data are average of the squared deviations of the highly skewed. II. Measures of Variability measurements about their mean m. The 1. Range: R = largest - smallest. variance of a sample of n measurements is 36 Key Concepts. 2. Variance a. the sum of the squared deviations of the Population of N measurements: b. Sample of measurements about their mean, divided by n measurements: 3. Standard deviation. (n – 1). 37 Key Concepts. IV. Measures of Relative 18 The Standard Deviation. In calculating Standing 1. pth percentile; p% of the the variance, we squared all of the measurements are smaller, and (100 - p)% deviations, and in doing so changed the are larger. 2. Lower quartile, Q 1; scale of the measurements. To return this position of Q 1 = .25(n +1) 3. Upper measure of variability to the original quartile, Q 3 ; position of Q 3 = .75(n units of measure, we calculate the +1) 4. Interquartile range: IQR = Q 3 - Q standard deviation, the positive square 1 V. Box Plots 1. Box plots are used for root of the variance. detecting outliers and shapes of 19 Two Ways to Calculate the Sample distributions. 2. Q 1 and Q 3 form the Variance. Use the Definition Formula: 5. ends of the box. The median line is in the -4. 16. 12. 3. 9. 6. -3. 9. 8. -1. 1. 14. interior of the box. 5. 25. Sum. 45. 0. 60. 38 Key Concepts. 3. Upper and lower 20 Two Ways to Calculate the Sample fences are used to find outliers. a. Lower Variance. Use the Calculational Formula: fence: Q 1 - 1.5(IQR) b. Outer fences: Q 3 5. 25. 12. 144. 6. 36. 8. 64. 14. 196. + 1.5(IQR) 4. Whiskers are connected to Sum. 45. 465. the smallest and largest measurements that 21 Some Notes. The value of s is ALWAYS are not outliers. 5. Skewed distributions positive. The larger the value of s2 or s, usually have a long whisker in the the larger the variability of the data direction of the skewness, and the median set. Why divide by n –1? The sample line is drawn away from the direction of standard deviation s is often used to the skewness. Statistics with Economics and Business Applications.ppt
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## Statistics with Economics and Business Applications

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