Descriptive Statistics

Chapter 3

Numerical Scales

Nominal scale-Uses numbers for identification (student ID numbers)

Ordinal scale- Uses numbers for ranking (order of winners in a race)

Interval scale- Uses numbers for ranking when units are equidistant (measuring temperature, but no absolute zero))

Ratio scale-This scale has qualities of equidistant units and absolute zero (age).

Descriptive Statistics

Descriptive Statistics- Statistics used to organize and describe data

Measures of Central Tendency- Statistic methods forobserving how data cluster around the mean

Normal Distribution-A symmetrical distribution with asingle numerical representation for mean, median, and mode

Measures of Dispersion-Statistical methods for observinghow data spread from the mean.

Counting the Data-Frequency

Look at the set of data that follows on the nextslide.

Each time a score occurred, a tally mark wasmade to count

Which number most likely represents the average score?

Which number is the most frequently occurring score?

Frequency Distribution

Scores1009998949089888275746860

Tally11111111111111 111111 11111111 111111

Frequency112257

1062111

AverageScore?

Most FrequentScore?

88

88

Tal

ly1 1 11 11 11

1111

11 1

111

11 1

111

1111

111 1 1 1

This frequency count represents data that closely represent a normal distribution.

Frequency Polygons

Data100 8999 8998 8998 8994 88 94 8890 7590 7590 7490 6890 60

5

4

3

2

1

60 68 74 75 88 89 90 94 98 99 100

Scores

Mean, Median, and Mode

Mean- The arithmetic average of a set of scores

Median-The middlemost point in a set of data

Mode- The most frequently occurring score in a set of data

Mean- To find the mean, simply add the scores and divide by the number of scores in the set of data.

98 + 94 + 88 + 75 = 355

Divide by the number of scores: 355/4 = 88.75

Median-The Middlemost point in a set of data

Data Set 110099999897969088858079

Data Set 2

100999897868278727068

Median96

The median is 84 for this set.84 represents

the middlemost point in

this set of data.

Mode-The most frequently occurring score in a set of data.

Find the modes for the following sets of data:

Data Set 3998989898975

Mode:89

Data set 499888887877270

88 and 87 are bothmodes for this set of data.This iscalled a bimodaldistribution. Threeor more modesrepresent a multi-modal distribution.

Measures of Dispersion

Range- Distance between the highest and lowest scores in a set of data.

100 - 65 = 35

35 is the range in this set of scores.

Variance-Describes the total amount that a set of scores varies from the mean.

1. Subtract the mean from each score.

When the mean for a set of data is 87, subtract87 from each score.

100 - 87 = 13 98- 87 = 11 95- 87 = 8 91- 87 = 4 85- 87 = -2 80- 87 = -7 60- 87 = -27

2. Next-Square each difference-multiply each difference by itself.

13 x 13 = 16911 x 11 = 1218 x 8 = 649 x 4 = 16-2 x -2 = 4 -7 x -7 = 49-27x -27 = + 729

3. Sum thesedifferences

1,152 Sum of squares

4. Divide the sum of squares by the number of scores.

1,152 divided by 7 =164.5714-This number represents the variance for this set of data.

5. To find the standard deviation, find the square root of the variance. For this set of data, find the square root of 164.5714.

The standard deviation for this set of dataIs 12.82 or 13.

Standard Deviation-Represents the typical amount that a score is expected to vary from the mean in a set of data.

Percentile Ranks -Scores that express the percentage of students who scored as well as or lower than a given student’s score.

A student received a percentile rank of 75 on a recent statewide exam. What does the 75 mean?

This indicates that 75% of the students in the same gradeor age group made the same score or a lower score. It alsomeans that 25% of this group made the same score or higher than this student.

Properties of a Normal Distribution orThe “Bell Curve”

The mean, median, and mode are represented by the same numericalvalue.Both sides of the curve are symmetricalAnything that occurs naturallyand is measurable will bedistributed in a normal distributionwhen sufficient data are collected.

The Normal Curve

Bell Curve

The Normal Distribution and Norm-Referenced Testing

Norm-referenced tests compare studentswith their age or grade peers.

Scores on these tests are compared with the nationalsample to determine if a specific student scoredabove or below the level expected for their ageor grade.

When a score is significantly above or below thelevel expected for the age or grade, it may indicatethat the student requires educational interventions.