Chapter 3-Central Tendency

Chapter 3 : 

Chapter 3 Central Tendency

Slide 3: 

Central tendency: A statistical measure to determine a single score that defines the center of the distribution. The goal of central tendency is to find the single score that is most typical or most representative of the entire group. The average Good to tell you some things Lubbock average income - $38,602 Miami average income – $28,999 Austin average income - $50,132 What do these tell us? Not tell us?

Slide 4: 

Three methods for dealing with central tendency, when looking at them keep in mind that we want to find the single most representative score The mean or arithmetic average is found by adding all the scores and then dividing them by number of scores Another way to think of the mean is the amount each individual would get if everything were added together and then divided up equally (communist!) The mean can also be seen as the balance point for a distribution

Mean Formulas : 

Mean Formulas

The Weighted Mean : 

The Weighted Mean You have 2 separate samples A) n=12 B) n=8 M=6 M=7 To calculate the overall mean we need: The overall sum of the scores for the combined group (∑X) The total number of scores in the combined group

The Weighted Mean cont. : 

The Weighted Mean cont. Find the sum of the scores for the first sample and the second sample and add them together For the total number of scores, add the n’s together So……

The Weighted Mean cont. : 

The Weighted Mean cont.

The Weighted Mean cont. : 

The Weighted Mean cont. Note that 6.4 is not halfway between the original samples, but is the mean for the entire group (one sample made a larger “contribution”

The Weighted Mean for a frequency distribution table : 

The Weighted Mean for a frequency distribution table

Characteristics of the Mean : 

Characteristics of the Mean Characteristics of the mean: Every score in the distribution contributes to the value of the mean, specifically: Every score adds to the total (∑X) Every score contributes one point to the number of scores (n) These two values (∑X and n) determine the mean Changing a score: Changing the value of any score will change the mean

The Mean cont. : 

The Mean cont. Introducing a new score or removing a score: Either of these will usually change the mean and require that a new computation be done However, if the new or removed score is exactly equal to the mean, then nothing will change – picture it like adding a block to the balance point, this does not shift the weight on either side

The Mean cont. : 

The Mean cont. Adding or subtracting a constant from each score: If a constant value is added to every score in a distribution, the same constant will be added to the mean, the same with subtraction For example, adding two points to every single score will add two points to the mean

The Mean cont. : 

The Mean cont. Multiplying or dividing by a constant: If every score is multiplies or divided by a constant value, the mean will change in the same way This is a common way to change the unit of measurement For example, to change from minutes to seconds you multiply by 60, to change from inches to feet you divide by 12 So if going from larger to smaller multiply, from smaller to larger, divide You just multiply the mean as well, if every score were multiplied by two, do the same to the mean

The Median : 

The Median The score that divides the distribution in half so that 50% of the individuals in a distribution have scores at or below the median No symbol, computations are same for sample or a population For 3, 5, 8, 10, 11 the media is 8 Exactly 8 When n is an even number you average the middle scores 3, 3, 4, 5, 7, 8 4 + 5 = 9/2 = 4.5

I didn’t enjoy this next part so here’s a funny picture first : 

I didn’t enjoy this next part so here’s a funny picture first

The Median cont. : 

The Median cont. In some situations involving a continuous variable, it is possible to divide a distribution in half by splitting one of the measurement categories into fractional parts so that exactly 50% of a population is above and below a specific point ex. N=10 1, 2, 2, 3, 4, 4, 4, 4, 4, 5 See figure 3.7 4 would not be the actual midpoint here because we don’t automatically see it as 3.5 to 4.5 as we should Using 3.7, count up to where you hit exactly five boxes You want to take exactly 1/5 of the boxes so that it makes up a full box on the graph 1/5 of the interval between 3.5 and 4.5 is .20, so 3.7 is what you want as your true median

The Median cont. : 

The Median cont. What to do when several scores are tied at the median Determine the real limits of the interval that contains the precise midpoint Count the number of scores (boxes) below the identified interval Find the number of additional scores needed to reach exactly 50% Calculate a fraction with the number of additional scores needed over the total number of scores in the interval Add the fraction to the lower real limit of the interval This is sensible for a continuous variable but will not work for a discrete variable – for a discrete variable just list the scores in order to find the middle

Mean versus Median : 

Mean versus Median A special note: The distances above the mean have the same total distances as the ones below it so the mean will always be located within the group of scores (there will be at least one score above and one below). You can have a distribution in which a vast majority of the scores are located on one side of the mean but a large number on the other side throws this off (an outlier) – look at income and Bill Gates. The median, on the other hand, is based where one half of the scores are on one side and one half on the other, the mean and the median use different definitions of the middle

Mean versus Median : 

Mean versus Median Consider this range of incomes $10,000 n=13 $10,000 The mean for this group of incomes is $346,153 – but $10,000 look at how many of the scores fall below what $20,000 should be the average. In a case like this, the median $20,000 is much more effective because it better approximates $30,000 the middle ($50,000) $50,000 $50,000 $50,000 $50,000 $100,000 $4,000,000

The Mode : 

The Mode In a frequency distribution, the mode is the score or category that has the greatest frequency No symbols, no difference between sample and population Useful measure of central tendency because it can be used to determine the typical or average value for any scale of measurement, including a nominal scale Look at 3.4 (p88) – what good is this information otherwise? You can’t average these or arrange them in any sort of order in order to get the median

The Mode cont. : 

The Mode cont. It is possible to have more than one mode (two scores with the same frequency) Two modes is bimodal, more than that is multimodal A bimodal distribution is often an indication that two distinct groups of individuals exist within a population Beiber versus Slipknot The term mode can be casually used to refer score with relatively high frequencies

Selecting a measure of central tendency : 

Selecting a measure of central tendency This depends on several factors, but note that you can often use all three and that the mean is the accepted one

When to use the median : 

When to use the median Extreme scores or skewed distributions: Think of our income example, this wouldn’t have been affected even if the $4 million had been $90 trillion Undetermined values: What if someone can’t solve a puzzle in the allotted time? Still needs to be counted but also needs to be separated from the rest to avoid skewing everything You can’t even compute the mean for 3.5

When to use the median cont. : 

When to use the median cont. Open-ended distributions When there is a category for “X or more” or “X or less” Think of those internet surveys: How many times a week do you throw up? 1 2 3 4 5 6 7 8 or more Once again, you cannot compute a mean

When to use the median cont. : 

When to use the median cont. Ordinal scale: The median is always appropriate and usually the preferred measure of central tendency for ordinal scale measures Remember, you can determine direction but not distance on an ordinal scale and distance is the primary thing in the mean

When to use the mode : 

When to use the mode Nominal Scales: As these can be differentiated only by name, this is a good measure because it doesn’t need all the numbers Discrete variables: Recall that discrete variables are those that exist only in whole, indivisible categories Think number of kids in a family or rooms in a house The mode always identifies the typical case, so you can avoid fractions of kids and rooms Describing shape: It’s easy to calculate so it can be tacked on for extra understanding Picture a test with a mean of 72 but a mode of 80 – most people made an 80, but maybe a few really bad scores pulled the average down

Presenting means and medians in graphs : 

Presenting means and medians in graphs The value here is to sow several means or median simultaneously It is customary to list the values for the different groups on the horizontal axis Typically these are the independent or quasi-independent variable Values for the dependent variable are on the vertical axis The means or medians are then listed using a histogram or bar graph

Central tendency and the shape of distribution : 

Central tendency and the shape of distribution There are times when the mean, median and mode will be similar, others when they will not – this is correlated with the shape of distribution Symmetrical Distribution: Two mirror sides The mean and the median will be the same and exactly in the center See figure 3.13 – p96

Central tendency and the shape of distribution : 

Central tendency and the shape of distribution Still predictable Positively skewed Peak will be the mode Median to the right of the mode Mean to the right of the median because of being influenced by outliers Negatively skewed Flipped

And we’re done : 

And we’re done