Showing Data Center and Spread

Transcription

1 Knowledge Article: Probability and Statistics Showing Data Center and Spread A. Measures of Central Tendency Central tendency is a loosely defined concept that has to do with the location of the center of a distribution. This section defines the three most common measures of central tendency: mean, median, and mode. Mean Mean is the most common measure of the center. The mean, also known as the arithmetic mean, is the average value of the items in a data set. For example, to calculate the mean weight of 50 people, add the 50 weights and divide the sum by 50. Median Median is the midpoint of a distribution: the same amount of data points are above the median as are below it. The median is generally a better measure of the center when there are extreme values, or outliers, because it is not affected by the precise numerical values of the outliers. Example 1 Data indicating the number of months that someone who has AIDS lives after taking a new antibody drug are (smallest to largest): 3, 4, 8, 8, 10, 11, 1, 13, 14, 15, 15, 16, 16, 17, 17, 18, 1,,, 4, 4, 5, 6, 6, 7, 7, 9, 9, 31, 3, 33, 33, 34, 34, 35, 37, 40, 44, 44, 47 Calculate the mean and the median. There are 40 values here. To find the mean, just add them all up and divide by 40: To find the median, first use the formula for the location. The location is n , where n is the number of data points. 1

2 Starting at the smallest value, the median is located between the 0th and 1st values (the two 4s): 3, 4, 8, 8, 10, 11, 1, 13, 14, 15, 15, 16, 16, 17, 17, 18, 1,,, 4, 4, 5, 6, 6, 7, 7, 9, 9, 31, 3, 33, 33, 34, 34, 35, 37, 40, 44, 44, median = = 4 Example Suppose that, in a small town of 50 people, one person earns $5,000,000 per year and the other 49 each earn $30,000. Which is the more accurate description of the center, the mean or the median? 5,000,000 (49 30,000) mean = 19, median = 30,000 The median is a more accurate description of the center than the mean because 49 of the values are 30,000 and one is 5,000,000. Here, 5,000,000 is an outlier, and 30,000 gives us a better sense of the center of the data. Mode The mode is the most frequent value found in the data set. If a data set has two distinct values that each occur the highest number of times, then the set is bimodal. Example1 Calculate the mode for these 0 exam scores: 50, 53, 59, 59, 63, 63, 7, 7, 7, 7, 7, 76, 78, 81, 83, 84, 84, 84, 90, 93 The most frequent score is 7, which occurs five times. Mode = 7. Example Calculate the mode for these 0 exam scores: 5, 55, 58, 59, 60, 60, 60, 6, 63, 69, 69, 71, 73, 73, 73, 76, 78, 78, 80, and 73 are both modes (each score occurs three times), and the set is bimodal.

3 B. Spread of Data The three main measures of the spread of a data set are range, interquartile range, and standard deviation. Each of these measures gives a glimpse of spread in data, while the three taken together give a more detailed look. Range Range is the simplest measure of variability to calculate, and one you have probably encountered many times in your life. The range is simply the highest score minus the lowest score. Here s a sample data set with 10 numbers: 99, 45, 3, 67, 45, 91, 8, 78, 6, 51. What is the range? The highest number is 99, and the lowest number is 3. Subtract the lowest from the highest, 99 3 = 76, to find the range: 76. Now consider the two quizzes shown in the figure. On quiz 1, the lowest score is 5 and the highest score is 9. Therefore, the range is 4. The range on quiz was larger: the lowest score was 4 and the highest score was 10. Therefore, the range is 6. Quiz Scores Quiz Scores Quiz 1 Quiz Interquartile Range The interquartile range (IQR) is the range of the middle 50% of the scores in a distribution. It is computed as IQR = 75th percentile 5th percentile. For quiz 1, the 75th percentile is 8 and the 5th percentile is 6. So, the interquartile range is. Quiz has greater spread. Its 75th percentile is 9, its 5th percentile is 5, and its interquartile range is 4. Example For the following 13 real estate prices, calculate the IQR. Prices are in dollars. (Source: San Jose Mercury News.) 3

4 389950, 30500, , , , , , , , 59000, , , Order the data from smallest to largest: , , 30500, , , , , 59000, , , , , median = 488,800 Since there are 13 data points, we ll take the average of the third and the fourth to find Q1, and we ll take the average of the 10th and the 11th to find Q3: 30, ,000 Q1 = 308, , ,000 Q 3 = 649,000 IQR = 649, ,750 = 340,50 Standard Deviation Standard deviation is a very common and useful measure of variability to report, but it is most appropriate to use when the distribution is normal (a bell-shaped curve) or approximately normal. In that case, you can calculate the percentage of the distribution within a given number of standard deviations away from the mean. For example, with a normal distribution, 68% of the distribution is within one standard deviation of the mean. Likewise, approximately 95% of the distribution is within two standard deviations of the mean, and approximately 99.7% of the distribution is within three standard deviations of the mean. Therefore, if you had a normal distribution with a mean of 50 and a standard deviation of 10, then 68% of the distribution would be between 40 (50 10) and 60 ( ). Similarly, about 95% of the distribution would be between 30 (50 0) and 70 (50 + 0), and about 99.7% of the distribution would be between 0 (50 30) and 80 ( ). The symbol for the population standard deviation is σ. 4

5 This figure shows two normal distributions. Both distributions have means of 50. The blue distribution has a standard deviation of 5. The red distribution has a standard deviation of 10. For the blue distribution, 68% of the distribution is between 45 and 55. For the red distribution, 68% is between 40 and 60. Some of the graphing and analysis tools used in this course will display the standard deviation of a data set. You will not be calculating the value of standard deviation in this lesson, but you should understand two things clearly: Standard deviation is a measure of spread, with larger values meaning larger spread. Standard deviation is most useful for a normal or near-normal data distribution. For non-normal distributions, the standard deviation has little reporting value. This knowledge article is adapted from the following sources: Illowsky, Barbara, and Susan Dean. "Collaborative Statistics." Connexions. June, 011. Lane, David M., Project Leader, Rice University. Online Statistics Education: A Multimedia Course of Study. 5