File Name: central tendency and dispersion .zip
The measures of central tendency are not adequate to describe data. Two data sets can have the same mean but they can be entirely different. Thus to describe data, one needs to know the extent of variability.
Measures of central tendency: categories or scores that describe what is "average" or "typical" of a given distribution. These include the mode, median and mean. Percentile: a score below which a specific percentage of a given distribution falls. Positively skewed distribution: a distribution with a handful of extremely large values.
Negatively skewed distribution: a distribution with a handful of extremely low values. Measures of variability: numbers that describe the diversity or dispersion in the distribution of a given variable. Box plot: a graphic representation of the range, interquartile range and median of a given variable.
The mode is the category with the greatest frequency or percentage. It is not the frequency itself. In other words, if someone asks you for the mode of the distribution shown below, the answer would be coconut, NOT It is possible to have more than one mode in a distribution. Such distributions are considered bimodal if there are two modes or multi-modal if there are more than two modes.
Distributions without a clear mode are said to be uniform. The mode is not particularly useful, but it is the only measure of central tendency we can use with nominal variables. You will find out why it is the only appropriate measure for nominal variables as we learn about the median and mean next. The median is the middlemost number. In other words, it's the number that divides the distribution exactly in half such that half the cases are above the median, and half are below. Conceptually, finding the median is fairly simple and entails only putting all of your observations in order from least to greatest and then finding whichever number falls in the middle.
Note that finding the median requires first ordering all of the observations from least to greatest. This is why the median is not an appropriate measure of central tendency for nominal variables, as nominal variables have no inherent order.
In practice, finding the median can be a bit more involved, especially if you have a large number of observations—see your textbook for an explanation of how to find the median in such situations. Some of you are probably already wondering, "What happens if you have an even number of cases?
There won't be a middle number then, right? If your dataset has an even number of cases, the median is the average of the two middlemost numbers. One of the median's advantages is that it is not sensitive to outliers. An outlier is an observation that lies an abnormal distance from other values in a sample. Observations that are significantly larger or smaller than the others in a sample can impact some statistical measures in such a way as to make them highly misleading, but the median is immune to them.
In other words, it doesn't matter if the biggest number is 20 or 20,; it still only counts as one number. Consider the following:. These two distributions have identical medians even though Distribution 2 has a very large outlier, which would end up skewing the mean pretty significantly, as we'll see in just a moment.
The mean is what people typically refer to as "the average". The mean takes into account the value of every observation and thus provides the most information of any measure of central tendency. Unlike the median, however, the mean is sensitive to outliers. In other words, one extraordinarily high or low value in your dataset can dramatically raise or lower the mean. The mean, often shown as an x or a y variable with a line over it pronounced either "x-bar" or "y-bar" , is the sum of all the scores divided by the total number of scores.
In statistical notation, we would write it out as follows:. In that equation, is the mean, X represents the value of each case and N is the total number of cases. The fact that calculating the mean requires addition and division is the very reason it can't be used with either nominal or ordinal variables. A percentile is a number below which a certain percent of the distribution falls. For example, if you score in the 90th percentile on a test, 90 percent of the students who took the test scored below you.
If you score in the 72nd percentile on a test, 72 percent of the students who took the test scored below you. If scored in the 5th percentile on a test, maybe that subject isn't for you. The median, you recall, falls at the 50th percentile.
Fifty percent of the observations fall below it. A symmetrical distribution is a distribution where the mean, median and mode are the same. A skewed distribution, on the other hand, is a distribution with extreme values on one side or the other that force the median away from the mean in one direction or the other.
If the mean is greater than the median, the distribution is said to be positively skewed. In other words, there is an extremely large value that is "pulling" the mean toward the upper end of the distribution. If the mean is smaller than the median, the distribution is said to be negatively skewed. In other words, there is an extremely small value that is "pulling" the mean toward the lower end of the distribution. Distributions of income are usually positively skewed thanks to the small number of people who make ungodly amounts of money.
Consider the admittedly dated case of Major League Soccer players as an extreme example. When trying to decide which measure of central tendency to use, you must consider both level of measurement and skew. This is not so much the case for nominal and ordinal variables.
If the variable is nominal, obviously the mode is the only measure of central tendency to use. If the variable is ordinal, the median is probably your best bet because it provides more information about the sample than the mode does.
If the distribution is symmetrical, the mean is the best measure of central tendency. If the distribution is skewed either positively or negatively, the median is more accurate. As an example of why the mean might not be the best measure of central tendency for a skewed distribution, consider the following passage from Charles Wheelan's Naked Statistics: Stripping the Dread from the Data :.
Bill Gates walks into the bar with a talking parrot perched on his shoulder. The parrot has nothing to do with the example, but it kind of spices things up. Obviously none of the original ten drinkers is any richer though it might be reasonable to expect Bill Gates to buy a round or two.
This isn't a bar where multimillionaires hang out; it's a bar where a bunch of guys with relatively low incomes happen to be sitting next to Bill Gates and his talking parrot. In addition to figuring out the measures of central tendency, we may need to summarize the amount of variability we have in our distribution.
In other words, we need to determine if the observations tend to cluster together or if they tend to be spread out. Consider the following example:. Sample 2 has no variability all scores are exactly the same , whereas Sample 1 has relatively more one case varies substantially from the other four.
In this course, we will be going over four measures of variability: the range, the inter-quartile range IQR , the variance and the standard deviation. The range is the difference between the highest and lowest scores in a data set and is the simplest measure of spread. We calculate range by subtracting the smallest value from the largest value.
As an example, let us consider the following data set:. The maximum value is 85 and the minimum value is Whilst using the range as a measure of variability doesn't tell us much, it does give us some information about how far apart the lowest and highest scores are. It basically means "quarter" or "fourth. Finding the quartiles of a distribution is as simple as breaking it up into fourths. Each fourth contains 25 percent of the total number of observations.
Quartiles divide a rank-ordered data set into four equal parts. The values that divide each part are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively. Q1 is the "middle" value in the first half of the rank-ordered data set. Q2 is the median value of the data set Q3 is the "middle" value of the second half of the rank-ordered data set Q4 would technically be the largest value in the dataset, but we ignore it when calculating the IQR we already dealt with it when we calculated the range.
Thus, the interquartile range is equal to Q3 minus Q1 or the 75th percentile minus the 25th percentile, if you prefer to think of it that way. As an example, consider the following numbers: 1, 3, 4, 5, 5, 6, 7, Q1 is the middle value in the first half of the data set. Q3 is the middle value in the second half of the data set. A box plot also known as a box and whisker plot splits the dataset into quartiles. The body of the boxplot consists of a "box" hence, the name , which goes from the first quartile Q1 to the third quartile Q3.
Within the box, a horizontal line is drawn at Q2, which denotes the median of the data set. Two vertical lines, known as whiskers, extend from the top and bottom of the box. The bottom whisker goes from Q1 to the smallest value in the data set, and the top whisker goes from Q3 to the largest value. Below is an example of a positively skewed box plot with the various components labeled. Outliers are values are extreme values that for one reason or another are excluded from the dataset.
If the data set includes one or more outliers, they are plotted separately as points on the chart. The above diagram has a few outliers at the bottom. The horizontal line that runs across the center of the box indicates where the median falls.
Additionally, boxplots display two common measures of the variability or spread in a data set: the range and the IQR. If you are interested in the spread of all the data, it is represented on a boxplot by the vertical distance between the smallest value and the largest value, including any outliers.
The middle half of a data set falls within the interquartile range. In a boxplot, the interquartile range is represented by the width of the box Q3 minus Q1. The variance is a measure of variability that represents on how far each observation falls from the mean of the distribution.
The first exercise focuses on the research design which is your plan of action that explains how you will try to answer your research questions. Exercises two through four focus on sampling, measurement, and data collection. The fifth exercise discusses hypotheses and hypothesis testing. The last eight exercises focus on data analysis. This data set is part of the collection at the Inter-university Consortium for Political and Social Research at the University of Michigan. This data set is freely available to the public and you do not have to be a member of the Consortium to use it. A weight variable is automatically applied to the data set so it better represents the population from which the sample was selected.
While measures of central tendency are used to estimate "normal" values of a dataset, measures of dispersion are important for describing the spread of the data, or its variation around a central value. Two distinct samples may have the same mean or median, but completely different levels of variability, or vice versa. A proper description of a set of data should include both of these characteristics. There are various methods that can be used to measure the dispersion of a dataset, each with its own set of advantages and disadvantages. Standard Deviations Away From Mean. Probability of Occurance.
A measure of central tendency is a number used to represent the center or middle of a set of data values. The mean, median, and mode are three commonly used.
Measures of central tendency: categories or scores that describe what is "average" or "typical" of a given distribution. These include the mode, median and mean. Percentile: a score below which a specific percentage of a given distribution falls. Positively skewed distribution: a distribution with a handful of extremely large values. Negatively skewed distribution: a distribution with a handful of extremely low values.
In statistics , a central tendency or measure of central tendency is a central or typical value for a probability distribution. Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late s.
At last he was doing it, clogging his senses. You love what we do together as much as I do. The musicians and champagne arrived as ordered, her wet clothes sucking at her body. In various queerly shaped, so Chloe would have the television to herself, a black silhouette framed by the firelit window.
Average: It is a value which is typical or representative of a set of data. Averages are also called Measures of Central Tendency. Simple to calculate.
Measures of Central Tendency. • a measure that tells us where the middle of a bunch of data lies. • most common are Mean, Median, and Mode. Mean.
In statistics , dispersion also called variability , scatter , or spread is the extent to which a distribution is stretched or squeezed. Dispersion is contrasted with location or central tendency , and together they are the most used properties of distributions. A measure of statistical dispersion is a nonnegative real number that is zero if all the data are the same and increases as the data become more diverse. Most measures of dispersion have the same units as the quantity being measured.
Central tendency is the middle point of a distribution and measures of central tendency means measuring sets of data in terms of the central location of the data in a data set. Accordingly, measures of central tendency include three important tools — mean average , median and mode. Measures of central tendency are generally calculated among ungrouped and grouped data and the formulae for the same would be accordingly different.
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