What happens to the mean and median if we add or multiply each observation in a data set by a constant? Show Consider for example if an instructor curves an exam by adding five points to each student’s score. What effect does this have on the mean and the median? The result of adding a constant to each value has the intended effect of altering the mean and median by the constant. For example, if in the above example where we have 10 aptitude scores, if 5 was added to each score the mean of this new data set would be 87.1 (the original mean of 82.1 plus 5) and the new median would be 86 (the original median of 81 plus 5). Similarly, if each observed data value was multiplied by a constant, the new mean and median would change by a factor of this constant. Returning to the 10 aptitude scores, if all of the original scores were doubled, the then the new mean and new median would be double the original mean and median. As we will learn shortly, the effect is not the same on the variance! Looking Ahead!Why would you want to know this? One reason, especially for those moving onward to more applied statistics (e.g. Regression, ANOVA), is the transforming data. For many applied statistical methods, a required assumption is that the data is normal, or very near bell-shaped. When the data is not normal, statisticians will transform the data using numerous techniques e.g. logarithmic transformation. We just need to remember the original data was transformed!! ShapeThe shape of the data helps us to determine the most appropriate measure of central tendency. The three most important descriptions of shape are Symmetric, Left-skewed, and Right-skewed. Skewness is a measure of the degree of asymmetry of the distribution. Symmetric
Mean = Median = Mode Symmetrical Left-Skewed or Skewed Left
Median Mean Mode Skewed to the left Right-skewed or Skewed Right
Median Mean Mode Skewed to the right Note! When one has very skewed data, it is better to use the median as measure of central tendency since the median is not much affected by extreme values. Chapter 6. Univariate descriptive statistics1. Compute the mode, median, and mean for the following four sets of numbers:
Use this set of numbers for the following questions:
2.Assume the numbers in the data are the answers you get when you ask people "How many magazines do you subscribe to?" What are the proper measures of central tendency and dispersion for this data? Calculate their values.
3. Assume the numbers in the data are the answers you get when you ask people "Name your favorite television program." Then you classify each program according to its thematic content. You use a system that has seven different classes (eg. 1=science fiction, 2=comedy, 3=romance, 4=adventure, 5=news, ....). The numbers in the data indicate which category their favorite programs fall into. What are the proper measures of central tendency and dispersion for this data?
4. Assume the numbers in the data are the answers you get when you ask people "What is your household's annual income? I'm going to read a list of possible ranges, and I want you to stop me when I read the range that describes your household's income." You then read the following list and record their answers:
What are the proper measures of central tendency and dispersion for this data? Calculate their values.
5. Below are the final exam scores in percentages for students in a course on postmodernist approaches to analysis of individual differences in skiing preferences.
a. Which of the measures of central tendency are the most and least appropriate for this data?
b. Which tell you more about the relative performance of males and females on the exam?
c. Discuss the benefits and drawbacks of each measure of central tendency for this data.
d. Compute the range, interquartile range, and standard deviation.
e. Discuss the benefits and drawbacks of each measure of dispersion for this data.
6. Use the table of random numbers (Table 7 in Appendix B) for this question. Use the last two digits of the 5-digit numbers. Starting at the top of the second column, scan down and mark the numbers that are between 10 and 29, including 10 and 29. Do this until you get a total of 15 numbers. Write these 15 two-digit numbers on a piece of
paper. Calculate the median, the mean, and the standard deviation for these numbers. Use the computational equation for standard deviation. 7. Analyze all four sets of numbers in Question 1 in terms of which of the measures of central tendency are the most and least appropriate. For each set of numbers, discuss the benefits and drawbacks of each measure of central tendency. 8. On a mid-term exam, the median score is 73 and the mean is 79. Which student's score is likely to be further away from the median — the one at the top of the class or the one at the bottom? Why?
9. If the standard deviation of a sample is 5.3,
10. Compute the standard deviation, range, and interquartile range for the following data:
11. Multiply each of the nine numbers in Question 11 a by a constant, say 0.4, and calculate the standard deviation. What is the effect on the standard deviation of multiplying the numbers by a constant? Try it with a different constant, say 1.3. What is the effect? What is the general pattern here?
12. Subtract a constant, say 50.0, from each of the nine numbers in Question 11, and calculate the standard deviation. What is the effect on the standard deviation of subtracting a constant? Try it with a different constant, say 63.89. What is the effect? What is the general pattern here?
13. What is the nature of the sample data if s = 0 and n = 75?
Which of the following is affected by extreme values in the data?Arithmetic mean refers to the average amount in a given group of data. It is defined as the summation of all the observation in the data which is divided by the number of observations in the data. Therefore, mean is affected by the extreme values because it includes all the data in a series. Was this answer helpful?
Which of the following measures is not affected by extreme values in the data quizlet?Median is not affected by extreme values because it is the middle value in a data set. The measure of central tendency that is more sensitive to outlier is the mean. Mean is directly affected by extreme values because it is the average of all values in a data set.
What measures are affected by extreme values?Arithmetic mean takes into account the value of all items (i.e. very large and very small) in a series. Thus, it is only arithmetic mean which is affected by extreme values in the series.
Which central measure is not affected by extreme values?The median is less affected by outliers and skewed data than the mean, and is usually the preferred measure of central tendency when the distribution is not symmetrical.
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