Basic Statistics Module 6 Activity 4

Overview In this presentation, you will learn about basic statistics. You will lean about the following topics: Measures of central tendency Measures of variability Distribution

Measures of central tendency

Measures of central tendency A measure of central tendency gives you information about the average or common score in a group of scores. There are three measures of central tendency: Mean Median Mode

Mean The mean is also known as the arithmetic average. To get the mean of a group of scores, you add all of the scores and divide by the number of scores. For example, imagine that you give 10 tests. The scores are: 9, 8, 8, 9, 10, 7, 7, 4, 5, 8. To get the mean you would add all of the scores and divide by 10. 9+8+8+9+10+7+7+4+5+8 = 75 75 / 10 = 7.5 The mean is 7.5

Median The median is the midpoint of the scores. To get the median, you arrange the scores in order from lowest to highest and find the middle score. If you have an even number of scores, the median is the midpoint between the two halves. For example, let’s find the median of our earlier scores: 9, 8, 8, 9, 10, 7, 7, 4, 5, 8. First, we re-arrange them: 4, 5, 7, 7, 8, 8, 8, 9, 9, 10 There’s an even number of scores, so our median is between the 8 and the 8. So, the median is 8.

Mode The mode is the most frequent score. For example, let’s find the mode of our earlier scores: 9, 8, 8, 9, 10, 7, 7, 4, 5, 8. The most frequent score is 8, so the mode is 8.

Measures of central tendency Mean, median, or mode? Calculating the mean takes into account each score, so really high or really low scores will affect the mean. Median is a counting average, so really high or really low scores will not affect it. Mode is simply the most frequent score, so it is the least reliable measure of central tendency.

Measures of central tendency Why would you use the mean, median, or mode? These three measures are quick and easy calculations to see how a student group performed on an assessment. They tell you information about the overall group of students.

Measures of variability

Measures of variability A measure of variability gives you information about how much scores spread out from the measure of central tendency. We will look at two measures of variability: Range Standard deviation

Range Range tells you the distance from the highest score to the lowest score. You get it by subtracting the lowest score from the highest score. For example, let’s go back to our earlier scores: 9, 8, 8, 9, 10, 7, 7, 4, 5, 8. The highest score is 10, and the lowest score is 4. 10 – 4 = 6. The range is 6.

Standard Deviation The standard deviation, or sd, is a measure of the variability of scores. To find the sd, there are five steps: Add all of the scores together. Take the result and multiple it by itself. Divide this number by the number of scores. Multiple each individual score by itself. Add these numbers together. Subtract the number from step 1 from the number in step 2. Divide the number from step three by the number of scores minus one. Find the square root of the number from step 4.

Standard deviation Let’s calculate the sd for our earlier set of scores: 9, 8, 8, 9, 10, 7, 7, 4, 5, 8. Step 1: Add all of the scores together. Take the result and multiple it by itself. Divide this number by the number of scores. 9+8+8+9+10+7+7+4+5+8 = 75 75*75 = 5,625 5,625 / 10 = 562.5

Standard deviation Step 2: Multiple each individual score by itself. Add these numbers together. 9*9 = 81 8*8 = 64 9*9 = 81 10*10 = 100 7*7 = 49 7*7 = 49 4*4 = 16 5*5 = 25 Sum = 593

Standard deviation Step 3: Subtract the number from step 1 from the number in step 2. 593 – 562.5 = 30.5

Standard deviation Step 4: Divide the number from step three by the number of scores minus one. 30.5 / 9 = 3.39

Standard deviation Step 5: Find the square root of the number from step 4. Square root of 3.39 = 1.84 The standard deviation of these scores: 9, 8, 8, 9, 10, 7, 7, 4, 5, 8 is 1.84.

Measures of variability The range of scores is a very simple measure of variability. The standard deviation is a more informative measure. The sd tells you how much, on average, scores are different from the mean.

Measures of variability A small range or sd tells you that the scores do not vary a lot. A large range of sd tells you that the scores are very different. For a classroom achievement test, we hope that the average score is high with a small sd. A small sd would indicate that students are performing similarly. For a proficiency placement test, we expect to see a variety of scores. For this type of test, a larger sd would be expected.

Measures of variability Why do we use the range and standard deviation? The mean median and mode tell you how the students did generally. The range and standard deviation give you a better understanding of how all of your students did. The range can show the difference between the high and low scoring students. The standard deviation helps you understand the variation between your students.

Distribution

Distribution The distribution is the shape of the scores. It is easiest to understand visually. Look at the image below. This may look familiar to you.

Distribution Across the bottom, you can see the grades on a test, from 50 to 100. Up the left side, you can see the number of students. You can see that 20 students received an 80 on the test, but only 3 received a 100.

Distribution We call this a normal distribution because if you draw a vertical line through the middle, the left side will be the same as the right side.

Distribution There are many different ways that scores can be distributed. The image on the left is positively skewed and the image on the right is negatively skewed.

Distribution The distribution of scores will depend on the test that we are giving. A norm-referenced test should have a relatively normal distribution. A criterion-referenced test will have either a positively or negatively skewed distribution. If it is a pre-test where students do not know the material, it should be positively skewed. If it is an achievement test, it should be negatively skewed.

Distribution Achievement test: negatively skewed, most of the scores are high. Pre-test: positively skewed, most of the scores are low.

Review: Mean Median Mode Standard deviation Normal distribution Key Terms & Concepts Review: Mean Median Mode Standard deviation Normal distribution

Next Steps Now you should read the Reliability PowerPoint presentation. This is activity #5.

Sources Carr, N. T. (2011). Designing and analyzing language tests. Oxford: Oxford University Press. Miller, M. D., Linn, R. L., & Gronlund, N. E. (2009). Measurement and assessment in teaching. New Jersey: Pearson.