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Interpreting Data Mean, Median, and Mode Understand the terms mean, median, mode, standard deviation Use these terms to interpret data collected during experiments Mean … the average number Median … the value that lies in the middle after organizing all the numbers Mode … the most frequently occurring number Which measure of Central Tendency should be used? The measure you choose should give you a good indication of the typical score in the sample or population. The mean (or average) is found by taking the sum of the numbers and then dividing by how many numbers you added together. The number that occurs most frequently is the mode. When the numbers are arranged in numerical order, the middle one is the median. 5/24/2017 The mean (or average) is found by adding all the numbers and then dividing by how many numbers you added together. Example: 3,4,5,6,7 3+4+5+6+7= 25 25 divided by 5 = 5 The mean is 5 5/24/2017 Notation x is pronounced ‘x-bar’ and denotes the mean of a set of sample numbers values x = x n means add or sum of a set of numbers x represents the individual data numbers n represents the how many data numbers are in a sample N represents the number of data numbers in a population 5/24/2017 Lowest 55 60 75 80 80 80 83 83 93 93 93 93 93 Highest First add all the grades together. The total equals 1061 Now divide 1061 by 13 (total grades in the class) Lowest 55 60 75 80 80 80 83 83 93 93 93 93 93 Highest 5/24/2017 First add all the grades together. The total equals 1061 Now divide 1061 by 13 (total grades in the class) The answer is 81.61 The mean is 81.61 Lowest 55 60 75 80 80 80 83 83 93 93 93 93 93 Highest 5/24/2017 Same thing, just fancy x is pronounced ‘x-bar’ and denotes the mean of a set of sample values x x = n µ is pronounced ‘mu’ and denotes the mean of all values µ = in a population x N Calculators can calculate the mean of data The mean is sensitive to every value, so one exceptional value can affect the mean dramatically. The median (next slide) overcomes that disadvantage. The number that occurs most frequently is the mode. Example: 2,2,2,4,5,6,7,7,7,7,8 The number that occurs most frequently is 7 The mode is 7 5/24/2017 The mode is the number which occurs most often. The number which occurs most often is 93 Lowest 55 60 75 80 80 80 83 83 93 93 93 93 93 Highest 5/24/2017 The mode is the number which occurs most often. The number which occurs most often is 93 The mode is 93 Lowest 55 60 75 80 80 80 83 83 93 93 93 93 93 Highest 5/24/2017 When numbers are arranged in numerical order, the middle one is the median. Example: 3,6,2,5,7 Arrange in order 2,3,5,6,7 The number in the middle is 5 The median is 5 5/24/2017 The median is the number in the middle of numbers which are in order from least to greatest. If we count from both sides the number in the middle is 83. Lowest 55 60 75 80 80 80 83 83 93 93 93 93 93 Highest 5/24/2017 The median is the number in the middle of numbers which are in order from least to greatest. If we count from both sides the number in the middle is 83. The median is 83 Lowest 55 60 75 80 80 80 83 83 93 93 93 93 93 Highest 5/24/2017 If these were your math grades, what would you learn by analyzing them? The mean was 81.61. In order to raise your grades, you would have to make higher than an 81.61 on the rest of your assignments. The mode was 93 which was your highest grade. You could look at these papers to see why you made this grade the most. The median is a 83. This means that most of your grades were higher than your average. Find your weak area and try to improve. 5/24/2017 Real Life ◦ Knowing the mean, median, and mode will help you better understand the scores on your report card. By analyzing the data (grades) you can find your average, the grade you received most often, and the grade in the middle of your subject area. ◦ Better understanding your grades may lead to better study habits. 5/24/2017 Mean … the most frequently used but is sensitive to extreme scores e.g. 1 2 3 4 5 6 7 8 9 10 Mean = 5.5 (median = 5.5) e.g. 1 2 3 4 5 6 7 8 9 20 Mean = 6.5 (median = 5.5) e.g. 1 2 3 4 5 6 7 8 9 100 Mean = 14.5 (median = 5.5) Median … is not sensitive to extreme scores … use it when you are unable to use the mean because of extreme scores Mode … does not involve any calculation or ordering of data … use it when you have categories (e.g. occupation) English Frequency 300 250 Mean: 54 200 Median: 56 150 Mode: 63 100 50 Mean = 53.78 Std. Dev. = 19.484 N = 4,253 0 0 20 40 60 English 80 100 In everyday life many variables such as height, weight, shoe size and exam marks all tend to be normally distributed, that is, they all tend to look like the following curve. Mean, Median, Mode 0.025 0.02 0.015 0.01 0.005 0 0 20 40 60 80 100 It is bell-shaped and symmetrical about the mean The mean, median and mode are equal It is a function of the mean and the standard deviation Measures that indicate the spread of scores: Range Standard Deviation Range It compares the minimum score with the maximum score Max score – Min score = Range It is a crude indication of the spread of the scores because it does not tell us much about the shape of the distribution and how much the scores vary from the mean Two distributions may have the same range but in one distribution the scores may be tightly packed around the mean while in the other they may be more spread out from the mean. Let’s determine the mean, mode and median for your individual data Then we will calculate the population data for the class! Standard Deviation It tells us what is happening between the minimum and maximum scores It tells us how much the scores in the data set vary around the mean It is useful when we need to compare groups using the same scale Mean 0.03 = 50 Std Dev = 15 0.025 0.02 0.015 34% 2% 0.01 0.005 34% 2% 14% 14% 0 0 10 20 30 40 50 60 70 80 90 100 scores 5 -3 0% 20 -2 2% 35 -1 16% 50 0 50% 65 +1 84% 80 +2 95 +3 sd 98% 100% rank School A 50 10 Mean S.d. School B 60 10 School C 70 10 0 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 20 40 60 80 100 120 School A 50 10 Mean S.d. School B 50 13 School C 50 16 0 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 20 40 60 80 100 120 National Mean 55 10 Mean S.d. School A 60 15 School B 40 15 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 -20 0 20 40 60 80 100 120 Best Measure of Center Advantages - Disadvantages Table 2-13 Definitions Symmetric Data is symmetric if the left half of its histogram is roughly a mirror of its right half. Skewed Data is skewed if it is not symmetric and if it extends more to one side than the other. Skewness Figure 2-13 (b) Mode = Mean = Median SYMMETRIC Skewness Figure 2-13 (b) Mode = Mean = Median SYMMETRIC Mean Mode Median Figure 2-13 (a) SKEWED LEFT (negatively) Skewness Figure 2-13 (b) Mode = Mean = Median SYMMETRIC Mean Mode Median Figure 2-13 (a) SKEWED LEFT (negatively) Mean Mode Median SKEWED RIGHT (positively) Figure 2-13 (c) The z-score is a conversion of the raw score into a standard score based on the mean and the standard deviation. z-score = Raw score – Mean Standard Deviation Example z-score Mean = 55 65 – 55 15 = 0.67 Standard Deviation = 15 Raw Score = 65 Use table provided to convert the z-score into a percentile. z-score = 0.67 Percentile = 74.86% (from table provided) Interpretation: 75% of the group scored below this score. Mean S.d. National Mean 55 10 School A 60 15 School B 40 15 0.045 Z-score for Mean of School A = (60 – 55)/10 = 0.2 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 A z-score of 0.2 is equivalent to a percentile of 57.93% on a national basis Z-score for Mean of School B = (40 – 55)/10 = -1.5 A z-score of –1.5 is equivalent to a percentile of -20 0 20 40 60 80 100 120 (100-93.32)%, that is, 6.68%!