Descriptive Statistics

Descriptive Statistics Chapter 3 – Organizing and Describing Data

Testing Leads to Measuring:Working with Scores • When we give a test in class – let’s say a math test – the first score we get is a RAW SCORE – THE TOTAL NUMBER CORRECT • We subtract the number of items missed from the number of items on the test. • If there were 25 items on the test and Sally missed 5, what is her raw score? • 25-5=20 • But what does that really mean to you or Sally?

Testing Leads to Measuring:Working with Scores • We have to convert Sally’s raw score of 20 to some scale to understand what it means. • If we say that Sally’s math test had 25 items and each item was assigned a value of 4 points each – what would the total point value be of the test? • 25 x 4 = 100 points • What would Sally’s raw score of 20 be on that scale? • 20 x 4 = 80 points

Psychometric Scales of Measurement 1. Nominal scale-Uses names or numbers for identification only – our class roster of names is a nominal scale (others = gender, social security numbers). 2. Ordinal scale- Uses numbers for ranking student performance from better to worse (such as percentiles).

Psychometric Scales of Measurement 3. Interval scale- Uses numbers for ranking only when numbers are equal distance apart (such as temperature, IQ scores, or exam grades) and has no absolute zero. 4. Ratio scale- Uses numbers for ranking when numbers are equal distance apart, but has an absolute zero (such as weight or pulse).

Important Definitions Mean, Median, and Mode Mean- The arithmetic average of a set of scores The formula used is Median-The middlemost point in a set of data Mode- The most frequently occurring score in a set of data

When we have collected performance information on students from tests, we have sets of scores we call data sets. What is the range of these scores (difference between highest and lowest scores) Range = 86 points Working with Data Sets Data Set 1 100 91 84 84 77 64 64 49 49 28 14

What can we do with these scores? Data Set 1 100 91 84 84 77 64 64 49 49 28 14 MEAN • To learn more about a student’s performance or the whole class we can summarize and interpret scores using descriptive statistics. • Mean (Calculate Average) • Median (middle score) • Mode (most frequent score) 64 MEDIAN 64 MODE 84,64, 49

What else can we do with these scores? Data Set 1 100 91 84 84 77 64 64 49 49 28 14 • Variance (only important in theory, but must be computed to get standard deviation) • Standard Deviation (Distance from the mean)

Variance – Describes the total amount that a set of scores varies from the mean. Data Set 1 100-64 = 91-64 = 84-64 = 84-64 = 77-64 = 64-64 = 64-64 = 49-64 = 49-64 = 28-64 = 14-64 = • Subtract the mean from each score. • Next-square each difference (multiply each difference by itself). • Add up all the squared differences • (Sum of Squares) • Divide by n the number of scores • You have the variance

2. Next-Square each difference- multiply each difference by itself. 36 x 36 = 27 X 27 = 20 x 20 = 20 x 20 = 13 x 13 = 0 x 0 = 0 x 0 = -15 x -15 = -15 x -15 = -36 x -36 = -50 x -50 = 3. Sum these squared differences 7240 Sum of squares

4. Divide the sum of squares by the number of scores n. 7240 divided by 11 = 658.18 This number represents the variance for this set of data. = 658.18

Standard Deviation-Represents the typical amount that a score is expected to vary from the mean in a 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 658.18 The standard deviation for this set of data is 25.65 SD

Now we can graphically represent our data on a distribution curve On the traditional “Bell Curve” the mean, median, and mode are represented by the same numerical value. Both sides of the curve are symmetrical

Properties of Area Under the Normal Distribution 34% . 34% 14% 14% 2% 2% -2SD -1SD MEAN +1SD +2SD

The Normal Distribution Curve can be Skewed Positive Skew More low scores – could mean the test was too ______. Negative Skew More high scores – could mean the test was too _______.

Leptokurtic and Platykurtic Curves

Our Test Data on a Distribution Data Set 1 100 91 84 84 77 64 64 49 49 28 14 -2 sd 14 -1 sd 39 Mean 64 + 1 sd 84 + 2 sd 114

Types of Scores • Standard Scores - Any derived (converted) score that has been standardized. • Standardized refers to making a set of scores fit standard values (i.e. pre-determined mean with a pre-determined standard deviation)

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 84 on a recent statewide exam. What does the 84 mean? This indicates that 84% of the students in the same grade or age group made the same score or a lower score. It also means that 16% of this group made the same score or higher than this student.

Using Percentile Scores in the Classroom • Verbal Labels - percentages used to facilitate instruction: • Defining Mastery – usually 90% + correct • Instructional Level – usually 85% to 95% correct • Frustration Level – less than 85% correct • Independent Level – more than 95% + correct

Standard Score ~ z-Scores Formula and Information • Mean of 0 • Standard Deviation of 1 • Any raw score can be converted to a z-score using this formula • Allows for comparison of different variables • Allows for comparison of different scores on different test types

Let’s Convert Some of Our Test Scores to Z-scores Data Set 1 100 91 84 84 77 64 64 49 49 28 14 64 MEAN 25.65 SD

Using Z-Scores to Compare Test Performance • If you write two exams, in Math and English, and get the following student’s scores: • Math 70 (class = 55, = 10) • English 60 (class = 50, = 5) • Which test mark represents the better performance (relative to the class)?

Z-score Example cont. • Math Grade: z = (70-55)/10 z = • English Grade: z = (60-50)/5 z =

The Normal Distribution and Norm-Referenced Testing Norm-referenced tests often compare students with their age or grade peers. Scores on these tests are compared with the national sample to determine if a specific student scored above or below the level expected for their age or grade. When a score is significantly above or below the level expected for the age or grade, it may indicate that the student requires educational interventions. However, we must be careful with age/grade scores. Why?

Problems with Age/Grade Equivalents • Individual performance cannot be truly assessed using age/grade equivalents because: • Can be misinterpreted because score obtained is general and not specific. Does not show that child’s true performance only how he or she compares to others. • Age/grade performance is not always directly measured for each age or grade level. It is sometimes estimated. • There is really no such thing as “Average” for a particular age or grade. • It is unethical to make a high-stakes placement decision such as Special Education Services based on age/grade scores.

Important Definitions • Descriptive Statistics- Statistics used to organize and • describe data • Measures of Central Tendency- Statistical methods for observing how data cluster around the mean (median and mode) • Normal Distribution-A symmetrical distribution with a single numerical representation for mean, median, and mode (Bell Curve) • Measures of Dispersion-Statistical methods for • observing how data spread from the mean (variance and standard deviation)

Creating box and whisker plots • http://teachertube.com/viewVideo.php?video_id=115265&title=Mr__Smith_s_Box_and_Whisker_Plot

Descriptive Statistics