Welcome to…

The Exciting World of Descriptive Statistics in Educational Assessment!

Numerical Scales

Type1. Nominal scale-Uses numbers for identification

2. Ordinal scale- Uses numbers for ranking

3. Interval scale- Uses numbers for ranking when units are equidistant

4. Ratio scale-This scale has qualities of equidistant units and absolute zero.

Descriptive Statistics- Statistics used to organize and describe data

Type

• Measures of Central Tendency- Statistic methods for observing how data cluster around the mean

• Normal Distribution-A symmetrical distribution with a single numerical representation for mean, median, and mode

• Measures of Dispersion-Statistical methods for observing how 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 it

•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?

Tally

1 1 11 11 1111

1111

11

1111

111

111

11 1

11 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

Freq

uenc

y

Measures of Central TendencyMean, 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 2100100999897868278727068

Median96

Median

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

Find the modes for the following sets of data:

Data Set 3998989898975

Mode: Data set 499888887877270

Mode:

Measures of Dispersion

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

100 - 65 = 35

RANGE

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 these

Sum of squares

VARIANCE (Continued)

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

_____divided by_____ = ______

VARIANCE (Final Step)

VARIANCE

1. To find the standard deviation, find the square root of the variance.

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

√_____ = ______Std. Deviation

1. find the deviation score (x-M)2. Divide deviation score by standard deviation

Z score- represents the score in terms of standard deviation units

(x-M)/SD = Z score

Properties of a Normal Distribution orThe “Bell Curve”

• The mean, median, and mode are represented by the same numerical value.

• Both sides of the curve are symmetrical

• Anything that occurs naturally and is measurable will be distributed in a normal distribution when sufficient data are collected.

How “normal” is the population?

Properties of a Skewed Distribution

Positively Skewed:

• More scores fall below the mean

• Median occurs below the mean

• Mode occurs below the median

Negatively Skewed:

• More scores fall above the mean

• Median occurs above the mean

• Mode occurs above the median

Watch out for extrem

e scores!

Z Scores- derived scores that are expressed in standard deviation units

A student received a Z score of 2 on a recent statewide exam. What does the 2 mean?

This indicates that the student’s score was 2 standard deviations above the mean for that test.