SACJ2
Measures of central tendency
Central tendency
Describe points around which the rest of the scores focus
Three measures
Mean
Median
Mode
Mode is considered the typical or more frequently occurring score in a distribution of score.
Median is considered the central score, or that point which divides a distribution into two equal parts, with 50% of the distribution on one side of the median and 50% on the other side.
Mean is the arithmetic average score of all scores in a set of scores.
Each has it’s own assumptions– it is important to know these assumptions in order to know when one is appropriate to report during data analysis.
2
Mode
The most common score
Can be used with variables at all three levels of measurement
Most often used with nominal level variables
Finding the Mode
Count the number of times each score occurred
The score that occurs most often is the mode
If the variable is presented in a frequency distribution, the mode is the largest category
If the variable is presented in a line chart, the mode is the highest peak
The mode
22, 23, 25, 25, 26, 26, 26, 27, 27, 28, 29, 30, 31, 32, 33, 35
22, 23, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 29, 30, 31, 33
22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 28, 29, 30, 35, 35
Can have multiple modes, but no more than 3
4
Mode for grouped data
| Sample of convicted murderers & Sentence received | |
| Years sentences | f |
| 55-59 | 11 |
| 50-54 | 7 |
| 45-49 | 10 |
| 40-44 | 15 |
| 35-39 | 10 |
| 30-34 | 9 |
| 25-29 | 5 |
| 20-24 | 3 |
| N=70 |
| Sample of convicted murderers & Sentence received | |
| Years sentences | f |
| 50-54 | 15 |
| 45-49 | 10 |
| 40-44 | 15 |
| 35-39 | 10 |
| 30-34 | 7 |
| 25-29 | 15 |
| 20-24 | 5 |
| N=77 |
The mode for grouped data is defined as the midpoint of the interval containing the most frequencies. 3 is the maximum number for grouped modes as well. 4 there is no mode
Use the midpoint of the interval as the mode year
Example: the interval 40-44 has the highest amount of murders so this interval midpoint is our mode (42 years). 5/2= 2.5; 39.5 + 2.5= 42
One mode= unimodal
Two modes= bimodal
Three modes= multimodal
Nominal level or higher
Most popular is not always the most central score. Can be very far away from the central tendency
Deviant scores or outliers– scores located in one extreme or another (small or large)
5
Limitations of Mode
Some distributions have no mode
Some distributions have multiple modes
The mode of an ordinal or interval-ratio level variable may not be central to the whole distribution
Median
Exact center of distribution of scores
The score of the middle case
Can be used with variables measured at the ordinal or interval-ratio levels
Cannot be used for nominal level variables
Finding the Median
Array the cases from low to high (or from high to low)
Locate the middle case
If N is odd: the median is the score of the middle case
If N is even: the median is the average of the scores of the two middle cases
The median
12, 15, 17
12, 15, 17, 19
12, 15, 17, 100
9, 12, 15, 17, 100
10, 35, 39, 43, 55, 220, 320, 480, 2,000,000
9, 12, 13, 15, 15, 15, 15, 15, 15, 17, 19, 20
Point that divides a distribution of scores into two equal parts.
In an array of an uneven number of scores, the central score becomes the median.
When there is an even number we can find a median, but this number is a theoretical point dividing a distribution
15
16
16
15
55
15
9
Median for grouped data
Mdn=+(fn/ff) (i)
| Satisfaction Score | ||
| Interval | f | cf |
| 175-179 | 4 | 111 |
| 170-174 | 6 | 107 |
| 165-169 | 3 | 101 |
| 160164 | 13 | 98 |
| 155-159 | 8 | 85 |
| 150-154 | 7 | 77 |
| 145-149 | 10 | 70 |
| 140-144 | 9 | 60 |
| 135-139 | 10 | 51 |
| 130-134 | 15 | 41 |
| 125-129 | 11 | 26 |
| 120-124 | 10 | 15 |
| 115-119 | 5 | 5 |
| N=111 |
LL= lower limit of the interval containing the number of frequencies we need to divide the total number of scores into two equal parts.
fn= the frequencies we need in the interval
ff= the frequencies found in the interval
I = the interval size
N/2= 111/2= 55.5
We must find the point with 55.5 scores on one side and 55.5 scores on the other side.
51 is a close are we can get to 55.5
Mdn=139.5+ 4.5/9 X 5
=139.5+22.5/9
=139.5 + 2.5
=142
Assumes data that can be measured at the ordinal scare or higher.
More stable measure of central tendency in the sense that it divides the scores in half.
10
Centiles, deciles, & quartiles
Centiles– divide distributions of scores into 1 % units
Deciles– divide distributions of scores into 10% units
Quartiles– divide distributions of scores into 25% units
50%= 5th decile and the 2nd quartile
75 centile is the point leaving 75% of all scores below it and 25% of scores above it
33 centile is the point leaving 33% of all scores below it and 67% of scores above it.
75% of scores form the 111 in the last chart we take (.75)(111)= 83.2 and from there we need to use the formula from last slide and plug in the amounts.
11
Mean
The average score
Requires variables measured at the interval-ratio level but is often used with ordinal-level variables
Cannot be used for nominal-level variables
The mean or arithmetic average, is by far the most commonly used measure of central tendency
Characteristics of the Mean
The mean “balances” out all of the scores in a distribution; all scores “cancel out” around the mean.
The mean is the point of minimized variation of the scores, “least squares principle”
The mean is affected by all scores; all scores are used in the calculation of the mean.
Strength - The mean uses all the available information from the variable
Weaknesses
The mean is affected by every score
If there are some very high or low scores (as with skewed distributions), the mean may be misleading
The Mean:
=
18, 19, 19, 20, 21, 21, 22, 25, 29, 32, 35, 37, 37, 38, 41, 41, 41, 43, 47, 49, 60
= the sum of the scores
N= the number of scores
695/21= 33.1
Replace the 60 with 600. What does the mean become?
(58.8)
Median is most appropriate when there are extreme scores or outliers.
15
| Number of IPV incidents from women with PTSD | |||
| Intervals | f | MP | (f)(MP) |
| 57-59 | 8 | 58 | 464 |
| 54-56 | 9 | 55 | 495 |
| 51-53 | 3 | 52 | 156 |
| 48-50 | 10 | 49 | 490 |
| 45-47 | 10 | 46 | 460 |
| 42-44 | 8 | 43 | 344 |
| 39-41 | 11 | 40 | 440 |
| 36-38 | 19 | 37 | 703 |
| 33-35 | 12 | 34 | 408 |
| 30-32 | 7 | 31 | 217 |
| 27-29 | 3 | 28 | 84 |
| 24-26 | 8 | 25 | 200 |
| 21-23 | 7 | 22 | 154 |
| N=115 | (N)(MP)=4,618 |
16
The mean for grouped data
=
=
| Number of IPV incidents from women with PTSD | |||
| Intervals | f | MP | (f)(MP) |
| 57-59 | 8 | 58 | 464 |
| 54-56 | 9 | 55 | 495 |
| 51-53 | 3 | 52 | 156 |
| 48-50 | 10 | 49 | 490 |
| 45-47 | 10 | 46 | 460 |
| 42-44 | 8 | 43 | 344 |
| 39-41 | 11 | 40 | 440 |
| 36-38 | 19 | 37 | 703 |
| 33-35 | 12 | 34 | 408 |
| 30-32 | 7 | 31 | 217 |
| 27-29 | 3 | 28 | 84 |
| 24-26 | 8 | 25 | 200 |
| 21-23 | 7 | 22 | 154 |
| N=115 | (N)(MP)=4,618 |
MP= interval midpoints
=
= = 40.1
17
Means, Medians, and Skew
When a distribution has a few very high or low scores, the mean will be pulled in the direction of the extreme scores
For a positive skew, the mean will be greater than the median
For a negative skew, the mean will be less than the median
When an interval-ratio level variable has a pronounced skew, the median may be the more trustworthy measure of central tendency