外文翻译原文名称：Fundamentals_of_StatisticsMeasures of Central Tendency and Location: mean, median, mode, percentiles, quartiles and deciles.x sorted x53 5355 5370 5358 5564 5757 5753 5869 6457 6868 6953 70The Measures of Central Tendency are Mean, Median and ModeMean →x-bar or x→ for a given variable, it is the sum of the values divided by the number of values (∑x i/n). In this case, we have n = 11. So we need to add all of the values together and divide by 11. ∑ = 657, x= 59.73Median→the number in a distribution of a variable’s response where one half of the values are above and one half of the values are below. To find the median, we first need to put our data in ascending order (smallest to largest). Then we can determine the me dian…if the value of n is odd, it issimply the middle observation, but if the value of n is even, it is the average of the two middle observations.In this case, n is odd, so the median will be the middle observation of our sorted values (the 6th value) (57)Mode→the value that occurs most frequently. If there are two different values most frequently occurring, the data are said to be bi-modal. If there are more than two modes, and the distribution is said to be multi-modal. In this case, the value that occurs most often is 53. So, the mode is 53.The measures of location are Percentile, Quartile and DecilePercentile → the p th percentile is a value such that at least p percent of the observations are less than or equal to this value and at least (100 – p) percent of the observations are greater than orequal to this value. To calculate percentiles, we use indices (i).i = (p/100) n for p1, p2, p3,…p99If the answer is a whole number (an integer), then i is the average of (P/100)n and 1+ (P/100)n.If the index number is not a whole number, we ALWAYS round up. The positionof the index is the next whole number (integer) greater than the computed index.For example:i(p50)= (50/100)11 = 5.5...this rounds up to 6So, we would count from the lowest value of the sorted data to the index number (6).Since the calculated i was not a whole number we had to round up to find the valuewhere at least 50% of the values are equal to or lower than this value and at least 50%are equal to or higher than this value. In this case, the value of the 50th percentileis the 6th value...57 … Does this look familiar? → The 50th percentile is the samething as the median.What does it tell us? In this distribution, AT LEAST 50% of the observations areLESS THAN OR EQUAL TO 57 AND AT LEAST 50% of the observations areGREATER THAN OR EQUAL TO 57.i(p80)= (80/100)11 = 8.8...this round up to 9. The 9th value is 68.Again, since the index number is not a whole number, we round up. So, we would count from the lowest value of the sorted data to the index number (9). In this case, the value of the80th percentile is 68.Since this dataset has 11 observations, we won’t have any instances where our calculated index number is a whole number. However, if we just remove our value of 70 and create anew distribution, we will be able to see an example...53 53 53 55 57 57 58 64 68 69i(p30)= (30/100)10 = 3...this is a whole number, so we must take the 3rd and 4th values and average them to find the 30th percentile. (53 + 55)/2 = 54So, the value of the 30th percentile is 54.Return to our original data distribution ...Quartiles –are special cases of percentiles…Q1 = P25, Q2 = P50, Q3 = P75,These three values divide the distribution into 4 equal quartersi(Q1)= (25/100)11 = 2.75...this rounds to 3, so Q1 is the 3rd value (53)i(Q2)= (50/100)11 = 5.5...this round to 6, so Q2 is the 6th value (57)i (Q3) = (75/100)11 = 8.25...this rounds to 9, so Q3 is the 9th value (64)Measures of Dispersion or V ariability : Range, interquartile range (IQR), variance, standard deviation and coefficient of variation.Range = This tells us how wide the span is from the maximum value to the minimum value. (Max –Min) = Range. In this instance, the range is 69 - 53 = 16.Interquartile Range (IQR) = This tells us how wide the span is in the middle 50% of the data. (Q3 – Q1) = IQR. In this case ... 64 – 53 = 11We use the formula:1)(2--∑n x x = s 2The variance for these data is 454.18. For our purposes here, the computation of variance is just a step towards the computation of the standard deviation.Sample standard deviation (s ) is the positive square root of the variance.74.642.45=+= sSo the formula for sample standard deviation is…1)(2--∑n x xPopulation Variance (σ2)→uses the same formula in the numerator, but N instead of n-1 in thedenominator. Since we rarely have information about the entire population, we almost always use the formula for sample variance, s 2.Population Standard Deviation: σ = 2σ…since we rarely have information from the entire population, we use the formula for sample standard deviation, s .Coefficient of Variation: 100⎪⎭⎫⎝⎛x s tells us what percent the sample standard deviation is of the sample meanThis number is “relative” and is only of use in comparing the distribution of two or more variables.Suppose I have two samples, and I want to know which sample has more variability…If both samples have the same mean, the one with the higher standard deviation will have the greater variability. However, if they have different means, I need to calculate the coefficient of variation to determine which one has the most variability. xbar = 458, s = 112 versus xbar = 687, s = 192Standardized Data and Detecting OutliersZ -score: z =sxx -The z-score tells us how many standard deviations a value is from the mean. We can look at a picture of what a z-score tells us. In the Normal Curve…the mean is at the highest point and the curve tails off symmetrically in both directions.The sign of the z-score tells us which direction the value is from the mean on the Normal Curve. Negative values will be to the left, and positive values will be to the right.Standardizing Scores:Standard Normal Curve …the mean is zero, and the standard deviation is 1. The distribution is bell-shaped and symmetrical. The area under the curve is 1, and the tails of the curve extend out infinitely. They never actually touch the horizontal axis. The highest point on the curve is at the meanReturn to our data …let’s calculate the z -scores for each of the values…Empirical Rule →used when the distribution is assumed to known to be approximately normal. → Approximately 68% of the values will fall within 1 sd of the mean → Approximately 95% of the values will fall within 2 sd of the mean→ Approximately 99.9% of the values will fall within 3 sd of the meanChebyshev’s Theorem → doesn’t require that the data have a normal distributionSays that at least (1 – 1/z2) values will fall within z standard deviations of the mean.1-1/12 = 0, 1-1/22 = .75, 1-1/32 = .88889, 1-1/42 = .9375, 1-1/52 = .96 →We can’t make any assumptions about the percent of values that are within 1 sd of the meanBut…→ At least 75% of the values will fall within 2 sd of the mean→ At least 88.9% of the values will fall within 3 sd of the meanWe use Chebyshev’s Theorem to estimate the variation in a distribution when→n < 30, or→ the shape of the distribution is unknown, or→the distribution is assumed to be non-normal.Outliers:suspect or extreme values of data that must be identified and scrutinized. If they are instances of incorrectly entered data, they should be corrected. If the value wasentered correctly and it is a valid number, it should remain in the dataset as part of the initial analysis.When we use the z-score method for identifying outliers, we assume that any value that has a z-score with an absolute value greater than 3.0 (that is less than -3.0 or greater than +3.0) is an outlier. Before we proceed with data analysis, we need to examine all outliers for accuracy. If we determine that the value is valid, we often run two sets of analysis. One with the outlier, and one without.Another way to identify outliers…Related to IQR is the Five number summary…minimum, Q1, Q2, Q3, & maximum. These values feed into upper and lower limits, and we graph them in a box plot.Use the box plot… The advantage of the boxplot is that it is not influenced by outliers or extreme values as are Z-scores.Box Plots –Whiskers show the range of data within the inner fences3(IQR)133(IQR) below Q1below Q1(IQR) above Q3above Q3 (Lower Outer & Inner Fences) (Upper Inner & OuterFences)Any values between the inner and outer fences are “unusual,” and any values out beyond the outer fences are “outliers.”Advantage of using the box plot method as well as the z-score method...the box plot method is not influenced by extreme values in the same way that the mean and the standard deviation are....it is saidto be a more conservative method of evaluating outliers.外文翻译原文课题名称：统计基础趋势和位置的划分： 意思是说，中位数，众数，百分位数，四分位数和十分位数。
