DESCRIPTIVE STATISTICS

Nothing new!! You are already using it!!

SIMPLE STATISTICS Mode Median Mean Standard deviation Frequency

M ODE 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

M EDIAN 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 What is the median? 7

M EAN 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 [( )/5)]

E XERCISE FOR MODE, MEDIAN, & MEAN Raw score Frequency ____ N=32 Mode? = 40 (6 students get this score)

E XERCISE FOR MODE, MEDIAN, & MEAN Raw score Frequency Top Bottom ____ 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 /2= 54

E XERCISE FOR MODE, MEDIAN, & MEAN Raw score Frequency ____ N=32 Mean? = 55 (total number of scores for all subjects is 1760 divided by 32 equals 55)

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.

W HY 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.

S AMPLE FREQUENCY TABLES Raw scoreFrequency ____ N=35 Raw scores (intervals of 5) Frequency N=35

SPSS OUTCOME OF FREQUENCY DATA 191 people completed the item Most thought the course was a little too hard.

Crosstabulation by totals Crosstabulation by row totals SPSS O UTCOME OF FREQUENCY DATA

S AMPLE F REQUENCY T ABLE (APA) MaleFemaleTotal Junior High School Teachers High School Teachers TOTAL

Table 1. Position, Gender, and Ethnicity of School Leaders AdministratorsTeachers Total ExperiencedInexp.ExperiencedInexp. Male Female Total More Complex Table (APA)

Another Sample Frequency Table Relationship between Self-Esteem and Gender Self-Esteem GenderLowMiddleHigh Male10155 Female51015

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

FREQUENCY POLYGON

Bar GraphPie Chart

W HICH SCALE OF DATA IS USED WITH WHICH ONE ? OK to compute.... Nomina l OrdinalIntervalRatio frequency distribution. Yes median and percentiles. NoYes add or subtract. No Yes mean, standard deviation, standard error of the mean. No Yes ratio, or coefficient of variation. No Yes

Exercises

Number of booksfrequency N= Twenty-five randomly selected students were asked the number of novels they have read this semester. The results are as follows:.

Number of booksfrequency N= 25 A) Find the mode. 1 (9 people read one book)

Number of booksfrequency N= 25 B) Find the median. 1 (mid point is 13; 5+9= 14)

Number of booksfrequency N= 25 C) Find the mean ( = 42÷25= 1.68)

Number of booksfrequency N= 25 D) What percentage of the students read fewer than 3 books. 80% (20x100÷25= 80)

2. F OLLOWING 3 STUDENTS ARE APPLYING TO THE SAME SCHOOL. T HEY CAME FROM DIFFERENT SCHOOLS WITH DIFFERENT GRADING SYSTEMS. W HICH STUDENT HAS THE BEST GPA WHEN COMPARED TO HIS SCHOOL ? St.GPASchool. Av. GPA s.d. A B C C