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Chapter 3 SUMMARIZING SCORES WITH MEASURES OF CENTRAL TENDENCY Moving Forward Your goals in this chapter are to learn: • What central tendency is • What the mean, median, and mode indicate and when each is appropriate • The uses of the mean • What deviations around the mean are • How to interpret and graph the results of an experiment New Symbols and Procedures • The symbol S is the Greek letter “S” and is called sigma. • This symbol means to sum (add). • You will see it used with a symbol for scores such as SX. This is pronounced as the “sum of X” and means to find the sum of the X scores. What is Central Tendency? Central Tendency • Measures of central tendency answer the question: – “Are the scores generally high scores or generally low scores?” What is Central Tendency? • A measure of central tendency is a statistic summarizing the location of a distribution on a variable • It indicates where the center of the distribution tends to be located Computing the Mean, Median, and Mode The Mode • The mode is the score having the highest frequency in the data • The mode is used to describe central tendency when the scores reflect a nominal scale of measurement Unimodal Distributions When a polygon has one hump (such as on the normal curve) the distribution is called unimodal. Bimodal Distributions When a distribution has two scores tied for the most frequently occurring score, it is called bimodal. The Median • The median is the score at the 50th percentile • The median is used to summarize ordinal or highly skewed interval or ratio scores Determining the Median • When data are normally distributed, the median is the same score as the mode • The symbol for the median is Mdn • Use the median for ordinal data or when you have interval or ratio scores in a very skewed distribution Determining the Median When data are normally distributed, the median is the same score as the mode. When data are only approximately normally distributed: 1. Arrange the scores from lowest to highest 2. Determine N 3. If N is an odd number, the median is the score in the middle position 4. If N is an even number, the median is the average of the two scores in the middle The Mean • The mean is the score located at the exact mathematical center of a distribution • The mean is used to summarize interval or ratio data in situations when the distribution is symmetrical and unimodal Computing a Sample Mean • The formula for the sample mean is SX X N Comparing the Mean, Median, and Mode • All three measures of central tendency are located at the same score on a perfectly normal distribution • In a roughly normal distribution, the mean, median, and mode will be close to the same score • The mean inaccurately describes a skewed distribution Central Tendency and Skewed Distributions Applying the Mean to Research Deviations • A score’s deviation is equal to the score minus the mean • In symbols, this is X X • The sum of the deviations around the mean is the sum of all the differences between the scores and the mean • It is symbolized by S X X Summarizing Research • Scores are from the dependent variable • Choose the mean, median, or mode based on 1. The scale of measurement used on the dependent variable and 2. The shape of the distribution, if you have interval or ratio scores Graphing the Results of an Experiment • Plot the independent variable on the X (horizontal) axis and the dependent variable on the Y (vertical) axis • Create a line graph when the independent variable is an interval or a ratio variable • Create a bar graph when the independent variable is a nominal or ordinal variable Line Graphs A line graph uses straight lines to connect adjacent data points Bar Graphs The bar above each condition on the X axis is placed to the height on the Y axis that corresponds to the mean score for that condition. Example For the following data set, find the mode, the median, and the mean. 14 14 13 15 11 15 13 10 12 13 14 13 14 15 17 14 14 15 Example—Mode • The mode is the most frequently occurring score • In this data set, the mode is 14 with a frequency of 6 Example—Median The median is the score at the 50th percentile. To find it, we must first place the scores in order from smallest to largest. 10 11 12 13 13 13 13 14 14 14 14 14 14 15 15 15 15 17 Example—Median • Since this data set has 18 observations, the median will be half-way between the 9th and 10th score in the ordered dataset • The 9th score is 14 and the 10th score also is 14. To find the midpoint, we use the formula: 14 14 14 2 • The median is 14 Example—Mean • For the mean, we need SX and N • We know N = 18 SX 246 SX 246 X 13.67 N 18