Statistics skewed distributions
Measures of relationships
- The degree of relationship between two variables is expressed as a correlation coefficient - Correlations tell the direction and the strength of the relationship. The closer to 1 (either a+1 or a-1, the stronger the correlation
skewed distribution positive skew
- The mean is pulled in the direction of the high scores (the tail to the right) - When a distribution is positively skewed, the mean is larger than the median
Negatively and directly related
- If two variables are negatively and directly related (as X increases, Y decreases, as X decreases, Y increases), the correlation coefficient will be close to -1.0, a perfect negative or inverse relationship - as I eat healthier, my body fat decreases - The older I get, the number of hairs on my head diminishes
Not related
- If two variables are not related, the correlation coefficient will be close to 0 - The relationship between IQ and shoe size
Degree or amount of skew
- Mean minus median- skew Ex: if the mean of a distribution is 200 and the median score is 190, the degree of skewness is 10 ( and in this case the distribution is positively skewed) 200-190= skew of 10 (positve)
Widely used correlations
- Pearson -> Used for interval or ratio measures -> The change has to be proportional -> Can I draw a straight line graph to best represent the data Spearman -> Used for ordinal data -> The change can be proportional but does not have to be -? "would a curved line be best to show the correlation?"
Positively and directly related
- if two variables are positively and directly related (as X increases, Y also increases or as X decreases, Y also decreases), the correlation coefficient will be close to +1.0, a perfect positive relationship - As children get older, their height and weight generally increase simultaneously - The more hours I provide in counseling, the more confident I feel in addressing my client's needs
skewed distribution
- not normal - not symmetrical and the values of the mean, the median, and the mode are different - The mean follows the direction of the tail - When picking a distribution that is more greatly skewed it doesn't matter whether the number is positive or negative (just pick the largest number)
skewed distribution negative
- the mean is pulled in the direction of low scores (the tail is to the left) = If the distribution is negatively skewed, the mean would be smaller than the median
Correlation coefficient
0.0 - .24= weak .25 -.74= Moderate .75 - 1.0 = Strong
Sample problem
What type of distribution do the following numbers represent: 11, 41, 23, 2, 30, 7, 18, 4, 12? a. A normal distribution b. a positively skewed distribution c. a negatively skewed distribution d. An inverted distribution Process #1: Calculate the mean or average Process #2: calculate the median or mid-pont Process # 3: Determine the answer - if both numbers (mean and median) are the same- you have a normal distribution (bell curve) - If the mean is smaller than the median, you have a negatively skewed distribution - If the mean is larger/bigger than the median, you have a positively skewed distribution
Example 2
if the mean of a distribution is 60 and the median score is 72, the degree of skewness is -12 (and is a negatively skewed distribution) 60-72= skew of -12