Organizing and Interpreting Test Data

• This lesson focuses on organizing and interpreting test data using tables and graphs

Learning Outcomes

• Organize test data using tables and graphs
• Interpret frequency distribution of test data

Performance Task

• Present test data in an organized manner, gathered from existing databases or pilot tests.
• Success is determined by developing data matrices, arranging data into frequency distributions (grouped), presenting graphical frequency distributions, interpreting published graphs, and applying these techniques to classroom assessment data.

Tables

• Tables and graphs help readers understand test results, which are useful for teachers, students, parents, administrators, and researchers.

Raw Data

• Raw scores are obtained from administering a test, questionnaire, or inventory rating scale.
• Raw scores are not interesting nor meaningful

Frequency Distribution Table

• Simple frequency distributions present an ordered arrangement of scores, which is better than a random list of raw scores

• Scores can be listed in ascending or descending order

• Created by tallying scores and recording the frequency of each score

• Frequency tables allow quick identification of the highest and lowest scores, and the counts for each score.

• Frequency and percent columns give specific information about number and percentage of students who got a test score

• The cumulative percentage calculates the percentage of cumulative frequency below a score in the dataset

Grouped Frequency Distribution

• Condenses data by grouping scores into class intervals
• Conventions:
• Class intervals should be equal (multiples of 5, 10, 100 are desirable)
• Formula to estimate class interval size, i = (H-L)/C, where H = highest score, L = lowest score, and C = number of classes.
• Conventional number of classes is between 7 and 20; an odd interval size resulting in whole number midpoints.
• Class intervals should start at a value which is a multiple of the class width
• Open-ended class intervals should be avoided.

Presenting Data Graphically

• Visual models can increase reader engagement
• Graphs are useful for comparing the test results of different groups

Histogram

• Appropriate for quantitative data (e.g., test scores).
• Columns representing class intervals; column height represents frequency (number of observations).
• Statistical software can construct histograms
• Focus on information extraction from a histogram.

Frequency Polygon

• Used for quantitative data; a line graph for presenting test scores.
• Lines connect data points, allowing comparison of data sets on one axes
• Can be manually constructed from a histogram by locating the midpoint on top of each histogram bar and connecting midpoints

Cumulative Frequency Polygon

• Plots cumulative frequencies
• Points plotted above the exact limits of the interval
• Useful for visualizing the number of observations below a certain score
• Cumulative percentage polygon is useful for comparing groups with unequal number of observations

Bar graph

• Frequencies in categories of a qualitative variable
• Columns represent categories, column height represents frequency
• Independent variable plotted on horizontal (x) axis, dependent variable on vertical (y) axis
• Bar graphs can be presented horizontally
• Good for comparison of test performance of groups categorized in two or more variables

Box-and-Whisker Plots

• Depicts the distribution of test scores through quartiles

• First quartile (Q1): the point below which 25% of scores lie

• Second quartile: the median (defines upper and lower 50% of scores)

• Third quartile (Q3): the point above which 25% of scores lie

• Shaded rectangle represents middle 50% of the data

• The line dividing the rectangle is the median

• Top side of rectangle is Q3, bottom side is Q1

• These plots help easily visualize score concentration, distribution into quartiles, and separation of quartiles.

• Whiskers extend from the box, representing the range for the bottom and top 25% data values, excluding outliers

• Outliers are extreme scores that can send an important message

Pie Graph

• Represents categorical data using a circle divided into slices
• Categories are represented by slices, sized according to the percentage in each category
• The size of the pie is determined by the percentage of students who belong in each category

Choosing a Graph

• Histogram is easiest for quantitative data.
• Bar graphs: Qualitative data; comparing subgroups
• Frequency/percentage polygons: Treating quantitative data.
• Cumulative frequency/percentage polygons: Determining distribution percentages
• Cumulative percentage polygon: Comparing groups with unequal numbers
• Box-and-whisker plots: Difficult to construct, but provide insightful test data information

Shapes of Frequency Distributions

• A frequency distribution arranges of a set of observations
• Variations in shapes of frequency distributions

Skewness

• Normal Distribution: symmetrical, bell-shaped; higher frequencies in center

• Half the area of the curve is a mirror reflection of the other half

• IQ scores, height, and weight often follow normal distribution

• Asymmetrical Distributions (skewed distributions):

• Degree of asymmetry called skewness.

• In negatively-skewed distributions: More high scorers, tail to the left (lower scores)

• In positively-skewed distributions: More low scores, tail to the right (higher scores)

• Rectangular Distribution: uniform distribution; each score or class interval has same frequency

Number of peaks

• Unimodal: One peak
• Bimodal: Two peaks
• Multimodal: More than two peaks
• Symmetry is not required for Unimodal, bimodal, or multimodal distributions

Kurtosis

• Flatness of the distribution, consequence of the height/peak
• Platykurtic: Flat, broad
• Mesokurtic: Normal distribution.
• Leptokurtic: Steep, slim