Descriptive Stats: Central Tendency PDF

Descrip(ve Stats: Central Tendency (Statistics that describe or summarize) What’s typical? http://www.youtube.com/watch?v=4B2xOvKFFz4 How useful are centers alone for conveying the true characteristics of a distribution? What is Central Tendency? A single value that represents the...

Descrip(ve Stats: Central Tendency (Statistics that describe or summarize) What’s typical? http://www.youtube.com/watch?v=4B2xOvKFFz4 How useful are centers alone for conveying the true characteristics of a distribution? What is Central Tendency? A single value that represents the d. y A de b rr ad de an u ivi center or typical value of a dataset ge Fin yo. D th d th rs er e e be th nu m m ge m id Summarizes complex data into one nu to be dl of rs rs e n nt be in um number ou um or b de er am n e he rb. th d t y Three main measures: si d A ze. Mean Fi d n er o um er. th fte b st b Median e n le num nu in n m th a t be e sm ges Mode r t se l r ha qu th la t o en ct he e cc ce t bt om ur. a S fr s r m os u t Central Tendency A measure of central tendency is a summary measure that attempts to describe a whole set of data with a single value that represents the middle or centre of its distribution. The goal of central tendency is to identify the single value that is the best representative for the entire set of data. There are three main measures of central tendency: the mode, the median and the mean. Each of these measures describes a different indication of the typical or central value in the distribution. 4 Central Tendency (cont.) Thus, central tendency (e.g., mean, median, mode) serves as a descriptive statistic because it allows researchers to describe or present a set of data in a very simplified, concise form. In addition, it is possible to compare two (or more) sets of data by simply comparing the average score (central tendency) for one set versus the average score for another set (for the same variable). 5 The Mean, the Median, and the Mode It is essenCal that central tendency be determined by an objecCve and well-defined procedure so that others will understand exactly how the "average" value was obtained and can replicate the process. No single procedure always produces a good, representa4ve value. Therefore, researchers have developed three commonly used techniques for measuring central tendency: the mean, the median, and the mode. 6 x̄ = 6.125 The Mean ArithmeCc average of all scores Formula: Popula>on: μ = ΣX / N (Greek) Sample: M = ΣX / n (Roman) Conceptual understanding: Balance point of distribu>on Equal distribu>on of total The Mean The mean is the most commonly used measure of central tendency. Computa2on of the mean requires scores that are numerical values measured on an interval or ra2o scale. The mean is obtained by compuCng the sum, or total, for the enCre set of scores, then dividing this sum by the number of scores. The mean of a populaCon is represented by the Greek leMer µ; the mean of a sample is represented by X 10 Calculating the Mean Calculate the mean of the following data: 1 5 4 3 2 Sum the scores (SX): 1 + 5 + 4 + 3 + 2 = 15 Divide the sum (SX = 15) by the number of scores (N = 5): 15 / 5 = 3 Mean = x̄ = 3 11 The Mean The sample mean, denoted as x̄, can be calculated as where x1, x2,..., xn represent the n (i.e., number of) observed values. The popula'on mean is also computed the same way but is denoted as µ. It is o@en not possible to calculate µ since populaBon data are rarely available. The sample mean is a sample sta's'c, and serves as an esBmate of the populaBon mean. This esBmate may not be perfect, but if the sample is good (representaBve of the populaBon), it is usually a preCy good esBmate. Changing the Mean Because the calcula2on of the mean involves every score in the distribu2on, changing the value of any score will change the value of the mean. Modifying a distribu;on by discarding scores or by adding new scores will typically change the value of the mean. To determine how the mean will be affected for any specific situaCon you must consider: 1) How the number of scores is affected, and 2) How the sum of the scores is affected. 14 Changing the Mean (cont.) If a constant value is added to every score in a distribuCon, then the same constant value is added to the mean. Also, if every score is mulCplied by a constant value, then the mean is also mulCplied by the same constant value. 15 When the Mean Won’t Work Although the mean is the most commonly used measure of central tendency, there are situaCons where the mean does not provide a good, representaCve value, and there are situaCons where you cannot compute a mean at all. When a distribuCon contains a few extreme scores (or is very skewed), the mean will be pulled toward the extremes (displaced toward the ‘tail’). In this case, the mean will not provide a "central" value. 16 x̄ = 6.125 When the Mean Won’t Work (cont.) With data from a nominal scale it is impossible to compute a mean, and when data are measured on an ordinal scale (ranks), it is usually inappropriate to compute a mean. Thus, the mean does not always work as a measure of central tendency and it is necessary to have alternaCve procedures available. 18 When To Use the Mean You should use the mean when… The data are interval or ra.o scaled Many people will use the mean with ordinally scaled data too … and the data are not skewed The mean is preferred because it is sensiCve to every score If you change one score in the data set, the mean will change 19 The Median The median is the value that splits the data in half when ordered in ascending order. If there are an even number of observa9ons, then the median is the average of the two values in the middle. Since the median is the midpoint of the data, 50% of the values are below it. Hence, it is also the 50th percentile. The Median If the scores in a distribuCon are listed in order from smallest to largest, the median is defined as the midpoint of the list. The median divides the scores so that 50% of the scores in the distribuCon have values that are equal to or less than the median. The median is simply another name for the 50th percen>le It is the score in the middle; half of the scores are larger than the median and half of the scores are smaller than the median ComputaCon of the median requires scores that can be placed in rank order (smallest to largest) and are measured on an ordinal, interval, or ra;o scale. 22 The Median (cont.) Usually, the median can be found by a simple coun;ng procedure: 1. With an odd number of scores, list the values in order, and the median is the middle score in the list. 2. With an even number of scores, list the values in order, and the median is half-way between the middle two scores. 23 The Median (cont.) One advantage of the median is that it is relaCvely unaffected by extreme scores. Thus, the median tends to stay in the "center" of the distribuCon even when there are a few extreme scores or when the distribuCon is very skewed. In these situa;ons, the median serves as a good alterna;ve to the mean. 24 Median Example What is the median of the following scores: 10 8 14 15 7 3 3 8 12 10 9 Sort the scores: 15 14 12 10 10 9 8 8 7 3 3 Determine the middle score: middle = (N + 1) / 2 = (11 + 1) / 2 = 6 Middle score = median = 9 26 When To Use the Median The median is ojen used when the distribuCon of scores is either posiCvely or negaCvely skewed The few really large scores (posi>vely skewed) or really small scores (nega>vely skewed) will not overly influence the median 27 The Mode 6 5 The mode is the score 4 Frequency that occurs most frequently in a set of 3 data 2 1 0 75 80 85 90 95 Score on Exam 1 28 The Mode The mode is defined as the most frequently occurring category or score in the distribution. In a frequency distribution graph, the mode is the category or score corresponding to the peak or high point of the distribution. The mode can be determined for data measured on any scale of measurement: nominal, ordinal, interval, or ratio. The mode is primarily used with nominally scaled data It is the only measure of central tendency that is appropriate for nominally scaled data The mode often is used as a supplemental measure of central tendency that is reported along with the mean or the median. 29 When To Use the Mode The mode is not a very useful measure of central tendency It is insensitive to large changes in the data set That is, two data sets that are very different from each other can have the same mode 7 120 6 100 5 80 4 60 3 40 2 1 20 0 0 30 1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90 100 Bimodal DistribuCons When a distribuCon 6 has two “modes,” it is 5 called bimodal 4 Frequency 3 2 1 0 75 80 85 90 95 Score on Exam 1 31 Bimodal Distributions It is possible for a distribuCon to have more than one mode. Such a distribuCon is called bimodal. (Note that a distribuCon can have only one mean and only one median.) In addiCon, the term "mode" is ojen used to describe a peak in a distribuCon that is not really the highest point. Thus, a distribuCon may have a major mode at the highest peak and a minor mode at a secondary peak in a different locaCon. 32 Multimodal Distributions If a distribuCon has 6 more than 2 “modes,” 5 it is called mul2modal 4 Frequency 3 2 1 0 75 80 85 90 95 Score on Exam 1 34 Selec7ng the Appropriate Measure: Central Tendency Consider: 1.Type of data (nominal, ordinal, interval, ra5o) 2.Distribu5on shape 3.Presence of extreme scores 4.Research ques5on Central Tendency and the Shape of the Distribution Because the mean, the median, and the mode are all measuring central tendency, the three measures are ojen systemaCcally related to each other. In a symmetrical distribuCon, for example, the mean and median will always be equal. or Right or Left 36 Examples in Behavioural Psychology Mean: Average reac)on )mes Test scores Physiological measurements Median: Likert scale responses Ordinal rankings Income data Mode: Preferred treatment methods Most common diagnoses Categorical survey responses Central Tendency and the Shape of the Distribu7on (cont.) If a symmetrical distribution has only one mode, then the mode, mean, and median will all have the same value. In a skewed distribution, the mode will be located at the peak on one side and the mean usually will be displaced toward the tail on the other side. The median is usually located between the mean and the mode. or Right or Le6 38 Reporting Central Tendency in Research Reports Repor;ng Central Tendency APA Style Guidelines: Mean: M = 10.3 Median: Mdn = 8.5 Mode: Typically reported in text Always include: Measure of variability (e.g., standard devia>on) Sample size Repor7ng Central Tendency in Research Reports In manuscripts and in published research reports, the sample mean is idenCfied with the leMer M. Median is represented as Mdn. There is no standardized notaCon for reporCng the mode. In research situaCons where several means are obtained for different groups or for different treatment condiCons, it is common to present all of the means in a single graph or table. 40