Chapter 4: Central Tendency

Formula for population mean

In this formula, the population mean is represented by a μ symbol and the total number of scores by a N, but the calculation is the same in that you are adding up (i.e.,∑ ) the scores (the Xs) and then dividing by the number of scores.

4.1 For which type of distribution (e.g., symmetrical, skewed) will the mean be a score in the middle of the distribution?

Symmetrical

Median

the middle score in a distribution; half the scores are above it and half are below it

Mode

the most frequently occurring score(s) in a distribution

How are the mean, median, and mode used as measures of central tendency?

The mean is the average, the median is the middle score, and the mode is the most common score. Each of these measures gives us summary value for the scores in a distribution, but in different ways.

Which measure of central tendency should I use when describing a distribution?

The mean is the most commonly used measure of central tendency; thus, it is often reported for comparison with other data sets. However, with skewed distributions, it is best to provide the median in addition to or instead of the mean. The mode is useful when describing data sets with many scores at the high and low ends of the scale or when reporting data on a nominal scale.

Formula for mean

The x̄ is a symbol that stands for the sample mean. The ∑ symbol (called a sigma) indicates that you should add up whatever comes after this symbol. In this case, you are adding the Xs, which means you should add up all the scores in the set of data. The n stands for the number of scores in the sample.

central tendency

Three main measures are mean, median, mode representation of a typical score in a distribution

For each set of scores below, calculate the mean by hand or with Excel or SPSS. How does the mean compare for these two sets of scores? Why do you think the mean is different for the two distributions? 50, 58, 63, 55, 52, 60, 54, 53, 61, 50 50, 58, 63, 55, 52, 60, 54, 53, 61, 96

55.6 60.2 The mean for the second set is nearly five points higher. The second set has an outlier at 96, which gives the distribution a positive skew.

You have designed a survey for a research study you are conducting as part of a course. You are interested in how many children people are interested in having in the future to see if this is related to how many siblings one grew up with. For each item, you ask people to respond with one of the following choices: 0 children/siblings, 1 child/sibling, 2 children/siblings, 3 children/siblings, and 4 or more children/siblings. Explain why the median or mode would be a better measure of central tendency for these data than the mean.

Because the last value is opened-ended, calculating a mean is not accurate (a value of 4 could mean 4 siblings, 5 siblings, 6 siblings, and so on). The mode would be best here, or you can use the median to show where the midpoint of the distribution is.

How do the mean, median, and mode compare for different distributions?

Because the mean is the average of the scores, it will be more influenced by extreme scores than the other measures. Thus, for a skewed distribution, the mean will be closest to the extreme scores, followed by the median and the mode, which will be closer to the middle of the distribution. For symmetrical distributions, however, the three measures will provide the same value that is in the middle of the distribution.

What can we learn about a distribution from measures of central tendency?

Central tendency measures provide a description of the typical score in a distribution.

For each measure described below, indicate which measure of central tendency you would choose and why. Figure 4.9 may help you decide by answering the questions in the chart that you are able to determine from the description of the measure. - speed to complete a Sudoku puzzle, measured in seconds - responses indicating which time of day (morning or afternoon) someone prefers to study for an exam -rating on a 1 to 5 scale indicating how pleasant someone found a social experience he or she is asked to participate in during a study

Median is best for a measure of speed because it is typically a skewed distribution. Time of day is a categorical variable here, so the mode is the best measure to use. Mean is likely to be used for rating scales, but if the distribution is skewed, the median can be used or if the distribution is bimodal (e.g., mostly 1s and 5s), the mode can be used.

For each data set below, calculate the median and mode. (Hint: Don't forget to put the scores in order from lowest to highest before you calculate the median.) How do these values compare for each data set—which one seems to be more representative of the scores in the data set? Explain your answer. 1 to 5 Ratings on Satisfaction with Courses in One's Chosen Major: 4, 2, 4, 3, 4, 4, 5, 4, 1, 4, 3, 3, 4, 5, 4, 5, 1 Accuracy on a Categorization Task (Percentage Correct): 78, 87, 90, 91, 75, 76, 88, 87, 77, 75, 92, 95, 78, 92, 87

Median: 4 Mode: 4 The median and mode scores are the same, so both are representative of the distribution. Median: 87 Mode: 87 The median and mode scores are the same, so both are representative of the distribution. (In both sets of numbers, the median and mode are the same.)

4.4 In your own words, explain why the median is a better measure of central tendency than the mean for distributions that contain extreme scores.

One or two outliers will tilt the mean toward that end, whether high or low. A median score gives a more accurate figure to associate with the average for a particular distribution of scores.

In your own words, explain why the median is a better measure of central tendency than the mean for distributions that contain extreme scores.

One or two outliers will tilt the mean toward that end, whether high or low. A median score gives a more accurate figure to associate with the average for a particular distribution of scores.

Outlier

an extreme deviation from the mean bias the mean toward the high or low end of the scale, depending on whether they are extremely high or extremely low. These distributions are then positively skewed (toward high scores) or negatively skewed (toward low scores). The smaller our sample size and the more extreme the outlier score is, the more these extreme scores will affect the mean in a skewed distribution.