CRIM 483 Descriptive Statistics

 Produces values that best represent an entire group of scores  Measures of central tendency—three types of information about the distribution of scores –Mean –Median –Mode

Mean  Mean=sum of all values divided by the total number of values __ X = (∑X)/n X = (∑X)/n  Where: __ __ –X = mean for group of scores –∑ = sum (sigma) –X = individual scores –n = number of scores (sample size)  The mean is typically the most central score

Calculating the Mean __ X = (∑X)/n X = (∑X)/n  Scores: 2,3,3,1,3,2,1,1  Mean=( )/8 =16/8 =16/8 =2 =2

Means, Cont’d.  Sensitive to extreme measures—can make the mean less representative of the scores and less useful as a measure of central tendency--potentially skews (or distorts) the results  Scores: 2,3,3,1,3,2,1,17  Mean=(∑ 2,3,3,1,3,2,1,17)/8 =32/8 =32/8 =4 =4

Weighted Means  Use weighted means when same values occur more than once –List all values –List the frequency with which each value occurs –Multiply the value * frequency –Sum all the value * frequency values –Divide by total frequency  See example on pg. 23

Median  Median=midpoint in a set of scores  Point at which 50% of the scores fall above and 50% of the scores fall below  To measure the median, the scores are placed in ascending numerical order and the value in the middle is selected –When there is an even number of values, compute the mean between the two values in the middle

Mean and Median Examples  Scores: 1,1,1,2,2,3,3,3 –Mean: 2 –Median: 2  Typically, the mean is considered the best overall measure unless there are extreme scores. In this case, the median is a better choice because it is not sensitive to extreme scores.  Scores: 1,1,2,2,3,3,3,17 –Mean: 4 –Median: 2.5

Mode  Mode=value that occurs most frequently among the scores  Mode is the value that occurred most often not the number of times  A distribution of values can be bimodal –Bimodal occurs when two values occur at the highest frequency Party Affiliation Number or Frequency Democrats90 Republicans70 Independents140 Hair Color Number or Frequency Red7 Blond7 Brown45 Black45

When to Use What  Depends on the type of data you are describing  Means are generally accepted as the best overall measure of scores  Medians are preferable when group of scores includes extreme values  Mean > precise than median > precise than mode