外文翻译原文名称：Fundamentals_of_Statistics Measures of Central Tendency and Location: mean, median, mode, percentiles, quartiles and deciles. x sorted x53535553705358556457575753586964576868695370The Measures of Central Tendency are Mean, Median and ModeMean x-bar or for a given variable, it is the sum of the values divided by the number of values (Sxi/n). In this case, we have n = 11. So we need to add all of the values together and divide by 11. S = 657, = 59.73Median the number in a distribution of a variables 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 medianif the value of n is odd, it is simply 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).57Mode 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 pth 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 or equal 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 position of 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 value where 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 percentile is the 6th value.57 Does this look familiar? The 50th percentile is the same thing as the median.What does it tell us? In this distribution, AT LEAST 50% of the observations are LESS THAN OR EQUAL TO 57 AND AT LEAST 50% of the observations are GREATER 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 the 80th percentile is 68.Since this dataset has 11 observations, we wont have any instances where our calculated index number is a whole number. However, if we just remove our value of 70 and create a new 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 percentilesQ1 = 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.57i(Q3) = (75/100)11 = 8.25.this rounds to 9, so Q3 is the 9th value.64Measures of Dispersion or Variability: 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 will use IQR in later processes, so we will want to keep this x(x-xbar)(x-xbar)253 53-6.73 -6.7345.29 45.2953 53-6.73 -6.7345.29 45.2953 53-6.73 -6.7345.29 45.2955 55-4.73 -4.7322.37 22.3757 57-2.73 -2.737.45 7.4557 57-2.73 -2.737.45 7.4558 58-1.73 -1.732.99 2.9964 644.27 4.2718.23 18.2368 688.27 8.2768.39 68.3969 699.27 9.2785.93 85.9370 7010.27 10.27105.47 105.47657 657-0.03 -0.03454.18 454.18657/11=59.73454.18/1045.2We use the formula: = s2The variance for these data is 454.18. For our purposes here, t