PSY 360 - Chapter 3: Central Tendency
What can change the mean?
1) Changing the value of any score will change the value of the mean. --> If N and/or SigmaX change, so will the mean. 2) If a constant value is added (or subtracted) to every score in a distribution, then the same constant value is added to the mean. 3) If every score is multiplied (or divided) by a constant value, then the mean is also multiplied by the same constant value.
When should you use median?
1) Extreme scores or skewed distributions. -> Mean 'pulled' toward extreme values. 2) Undetermined values. -> Impossible to determine mean. 3) Open-ended distributions. -> No upper (or lower) limit. Ex:Top category could be "5 or more" 4) Ordinal scale. -We know direction, but impossible to determine distance. -> It's a good choice because half of the scores are above the median and half of the scores are below the median.
How to find the median when N is even?
1) List the scores in order lowest to highest. 2) Find the point halfway between the middle two scores. Ex: 6, 8, 10, 12, 15, 20 ^ = 11 *The technique of listing and counting scores works well for finding the median of most distributions, especially for discrete variables.
How to find the median when N is odd?
1) List the scores in order lowest to highest. 2) Median is the middle score in the list. Ex: 5, 6, 6, 7, 8 ^ *The technique of listing and counting scores works well for finding the median of most distributions, especially for discrete variables.
What are the three different ways to measure central tendency?
1) Mean 2) Median 3) Mode *These are all computed in different ways and have different characteristics. Some of these word better than others in certain situations.
When should you use mean?
1) Mean is usually the preferred measure of central tendency. 2) It uses every score of the distribution. ->So, it produces a good representative value. 3) It closely related to variance and standard deviation. -> So, it is a valuable measure for inferential statistics. 4) However, in some situations, it is either impossible to compute or not very representative.
When should you use mode?
1) Nominal scale. - Does not measure quantity (distance or direction) - Impossible to compute mean or median. 2) Discrete variables. -Exist in whole, indivisible categories -Ex: average family having 2.4 children 3) Describing shape. - Requires little or no calculation -Exam score having mean of 72 and mode of 80. *Discrete variables - these are usually numerical variables. You can compute a mean but sometimes the mean is a fractional value that can't actually exist. You could get a mean of 2.4 children. This is true for median also. The mode gives you whole number values.
What are the two values needed to calculate the overall mean?
1) The overall sum of the scores for the combined group. 2) The total number of scores in the combined group.
What is a bimodal distribution?
A bimodal distribution is possibility for a distribution to have more than one mode. The mode is often used to describe a peak in a distribution that is not really the highest point. Thus, a distribution may have a major mode at the highest peak and a minor mode at a secondary peak. *A distribution can have only one mean and only one median. **There could also be no mode.
What is central tendency?
Central tendency is a descriptive statistical measure that identifies the center of distribution. Central tendency is define as a statistical measure to determine a single score that defines the center of a distribution.
What is central tendency's goal?
Central tendency's goal is to identify the single value that is the best representative for the entire set of data. Ex: Seattle, WA -> 53 degrees (average yearly temp.) --> 34 inches (average annual rain)
What are the definitions of mean?
Conceptually, the mean can also be defined in two ways: 1) The mean is the amount that each individual receives when the total (SigmaX) is divided equally among all individuals (N). 2) The mean is the balance point of the distribution because the sum of the distances below the mean is exactly equal to the sum of the distances above the mean.
How to find the precise mean?
If the scores are measurements of a continuous variable, you can find the median by placing scores in a histogram. Then, draw a vertical line through the distribution so that exactly half of the boxes are on each side of the line.
What is the central tendency and shape of the distribution if it was a skew distribution?
In a skewed distribution, the mode will be located at the peak on one side. The mean will usually be displace towards the tail on the other side. The median is usually located between the mean and the mode. There are two types of skewed distribution: positive and negative.
What is the central tendency and shape of the distribution if it is symmetrical, unimodal?
It is one peak with equal left and right so mean, median, and mode are the same.
What is the central tendency and shape of the distribution if it is symmetrical, bimodal?
It would have two peaks with a dip between the peaks where the median and mean are the same.
What is the formula for weighted mean?
M = (SigmaX1 +SigmaX2)/(n1 + n2) *Overall mean isn't located halfway between the two means. That would only be the case if the two means and n values were the same for both groups.
Median Vs. Mean
Mean and median both find the center of the distribution, but the center is defined differently. --> Mean - uses distance to find the center of the distribution (balance point) --> Median - uses the scores to find the center of the distribution - The distances above the mean must have the same total as the distances below the mean, balance point. -One advantage of the median is that it is relatively unaffected by extreme scores. The mean is pulled toward these extremes. - Thus, the median tends to stay in the "center" of the distribution even when there are a few extreme scores or when the distribution is very skewed. In these situations, the median serves as a good alternative to the mean.
How do you calculate mean?
Mean is the sum of scores (x) divided by total scores (N). Mean = SigmaX/N Scores need to be measured on an interval or ratio scale. You couldn't average scores on a nominal or ordinal scale of measurement.
Bottom Line Of Measuring Central Tendency
Most of the time, we can compute 2 or 3 measures of central tendency for the same set of data. --> They are usually similar, but can be very different.
What are the formulas for population mean and sample mean?
Population Mean: mu = SigmaX/N Sample Mean: M = SigmaX/n
What is an example of weighted mean?
Sample 1: n1 = 12, SigmaX1 = 72, M1 =6 Sample 2: n2 = 8, SigmaX2 = 56, M2 =7 Overall Mean: M = (72 + 56)/(12+8) = (128/20) =6.4 Overall mean isn't just the average of the two M values. One mean has more of a weight than the other, 6 is closer to 6.4 than 7.
When should you not use the mean?
The mean shouldn't be used if: 1) the distribution contains a few really extreme scores 2) the distribution is skewed - the mean is going to be pulled towards the extreme values and won't really provide a "central" value 3) the mean is from nominal or ordinal data
What is the median?
The median is midpoint of the list if the scores in a distribution are listed in order from smallest to largest. Exactly 50% of the individuals have scores at or below the median. Precise median = 50th percentile Scores must be on an ordinal, interval, or ratio scale.
What is the mode?
The mode is define as the most frequently occurring score or category. 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. --> MODE IS THE ONLY MEASUREMENT THAT CAN BE USED WITH NOMINAL!
What is weighted mean?
The weight mean (aka overall mean) combines 2 sets of scores and finds the overall mean for the combined group.
What is an example of precise median?
We can find the PRECISE median so that exactly 50% of the distribution is located below (or above) a certain point. 1, 2, 3, 4, 4, 4, 4, 6 (N=8) If we were dealing with a discrete variable, the median would be 4. BUT, the variable in this example is continuous, and 4 actually corresponds to the INTERVAL of 3.5-4.5, and median is a point within this interval. Since our N is 8, the median will have exactly 4 boxes (50%) on each side it. Count the boxes on the scale of measurement from left to right, once we get to 3.5 on the x-axis, we need 1 more box to reach 4 (50%). In this example, we need 1 out of the 4 boxes that occupy the space of the interval (3.5-4.5). So we need ¼ of the boxes that occupy a space of 1.00. ¼ of 1.00 is .25. 3.5 + .25 = 3.75
