DESCRIPTIVE STATISTICS

Nothing new!! You are already using it!!

SIMPLE STATISTICS Mode Median Mean Standard deviation Frequency

Mode The score attained by more students than any other scores. E.g. 25, 20, 18, 18, 18, 16, 15, 14, 14, 10, 10 What is the mode in this distribution? 18

Median The point which separates the higher half of a sample from the lower half (the midpoint) E.g. Uneven number of scores: 5, 4, 3, 2, 1 What is the median? 3 Even number of scores: 10, 8, 6, 4 7

Mean It is adding up all the scores and then dividing the sum by the total number of scores. 10, 15, 18, 23, 29= mean is 19 [(10+15+18+23+29)/5)]

Exercise for mode, median, & mean Raw score Frequency 78 75 70 67 66 62 59 55 53 51 49 40 32 1 2 3 4 6 ____ N=32 Mode? = 40 (6 students get this score)

Exercise for mode, median, & mean Raw score Frequency 78 75 70 67 66 62 59 55 -------- 53 51 49 40 32 1 2 3 2 Top 16 4 4 Bottom 16 6 ____ N=32 Median? = 54 (sum of the middle scores divided by 2) Total of 32 students middle score for the top 16 is 55 middle score for the bottom 16 is 53 55+53/2= 54

Exercise for mode, median, & mean Raw score Frequency 78 75 70 67 66 62 59 55 53 51 49 40 32 1 2 3 4 6 ____ N=32 Mean? = 54.75 (total number of scores for all subjects is 1752 divided by 32 equals 54.75)

STANDARD DEVIATION Averages are useful statistics but not sufficient. Eg. A: 19, 20, 25, 32, 39 (mean= 25; median=25) B: 2, 3, 25, 30, 75 (mean= 25; median=25) In both the mean and the median is 25; however, in A the scores are clustered around the mean and in B they are spread out. Thus, there is a need to describe the spread (variability) within a distribution.

There are various ways to measure the spread but the most useful one is standard deviation. The more spread out the scores are, the larger the deviation is, and the closer the scores are to the mean, the smaller the deviation. Thus, a sd. of 2.7 means there is less variability than a sd. of 8.3.

Why is standard deviation useful? E.g. If you are comparing test scores for different schools, the standard deviation will tell you how diverse the test scores are for each school. Imagine School A has a higher test score mean than School B. Your first reaction might be to say that the kids at School A are smarter. But a bigger standard deviation for one school tells you that there are relatively more kids at that school scoring toward one extreme or the other. Then, it is possible that lots of gifted students were sent to School A. In this way, looking at the standard deviation can point you in the right direction when asking why the information is the way it is.

FREQUENCY Giving raw scores and percentages. Quantitative data can be summarized using frequency tables.

Sample frequency tables Raw score Frequency 78 75 70 67 66 62 59 55 53 51 49 40 32 1 2 3 4 5 ____ N=35 Raw scores (intervals of 5) Frequency 76-80 71-75 66-70 61-65 56-60 51-55 46-50 41-45 36-40 31-35 1 2 7 4 8 5 3 --- N=35

SPSS outcome of frequency data 191 people completed the item Most thought the course was a little too hard.

SPSS Outcome of frequency data Crosstabulation by totals Crosstabulation by row totals SPSS Outcome of frequency data

Sample Frequency Table (APA) Male Female Total Junior High School Teachers 40 60 100 High School Teachers TOTAL 200

More Complex Table (APA) Table 1. Position, Gender, and Ethnicity of School Leaders Administrators Teachers Total Experienced Inexp. Male 50 20 150 80 300 Female 10 120 70 30 200 600

Self-Esteem Low Middle High Another Sample Frequency Table Relationship between Self-Esteem and Gender Self-Esteem Gender Low Middle High Male 10 15 5 Female

It is also possible to present this data in a graph, frequency polygon, bar chart, pie chart, etc.

FREQUENCY POLYGON

Bar Graph Pie Chart

Which scale of data is used with which one? OK to compute.... Nominal Ordinal Interval Ratio frequency distribution. Yes median and percentiles. No add or subtract. mean, standard deviation, standard error of the mean. ratio, or coefficient of variation.

Exercises

. 1. Twenty-five randomly selected students were asked the number of novels they have read this semester. The results are as follows: Number of books frequency 5 1 9 2 6 3 4 N= 25

A) Find the mode Number of books frequency 5 1 9 2 6 3 4 N= 25 . A) Find the mode Number of books frequency 5 1 9 2 6 3 4 N= 25 1 (9 people read one book)

B) Find the median Number of books frequency 5 1 9 2 6 3 4 N= 25 . B) Find the median Number of books frequency 5 1 9 2 6 3 4 N= 25 1 (mid point is 13; 5+9= 14)

C) Find the mean of the books that have been read Number of books . C) Find the mean of the books that have been read Number of books frequency 5 1 9 2 6 3 4 N= 25 1.48 (0+9+12+12+4= 37÷25= 1.48)

. D) What is the percentage of the students who have read fewer than 3 books? Number of books frequency 5 1 9 2 6 3 4 N= 25 20 students (5+9+6) out of 25 80% (20x100÷25= 80)

2. Following 3 students are applying to the same school 2. Following 3 students are applying to the same school. They came from different schools with different grading systems. Which student has the best GPA when compared to his school? St. GPA School. Av. GPA s.d. A 2.7 3.2 0.8 B 87 75 20 C 8.6 8 0.4 C