Statistics Used In Special Education National Association Of Special Education Teachers

Definition Statistics-Mathematical procedures used to describe and summarize samples of data in a meaningful fashion

Basic Statistics You Need to Understand Measures of Central Tendency Frequency Distributions Range Standard Deviation Normal Curve Percentile Ranks Standard Scores Scaled Scores T Scores Stanines

Measures of Central Tendency Measures of Central Tendency-A single number that tells you the overall characteristics of a set of data Mean Median Mode

Mean Definition: The Mean is the Mathematical Average It is defined as the summation (addition) of all the scores in your distribution divided by the total number of scores Statistically, it is represented by M

In the distribution: 8, 10, 8, 14, and 40, What is the Mean? Example Mean Problem In the distribution: 8, 10, 8, 14, and 40, What is the Mean?

Answer to Mean Problem The Mean is 16. Add up the scores: 8 +10+8+14+40 = 80 Adding the scores up gives you a total of 80. There are 5 scores 80/5 is 16 M = 16 The Mean is 16.

Problem with the Mean Score Extreme Scores can greatly affect the Mean In our example, the mean is 16 but there is only one score that is greater than 16 (The 40) So, extreme scores (whether high or low) can affect the Mean

Median Definition: The Median is the Midpoint in the Distribution It is the MIDDLE score Half the scores fall ABOVE the Median and half the scores fall BELOW the Median

Calculate the Median 8, 10, 8, 14, 40 In the distribution of scores:

Remember the Rule for Median Score **RULE: In order to calculate the Median, you must first put the scores in order from lowest to highest For our example, this would be 8, 8, 10, 14, 40

Answer to Median Problem 8, 8, 10, 14, 40 Cross of the low then cross off the high (in our example 8 & 40) Repeat until a Middle Number Obtained The Median is 10

What if There are Two Middle Scores? Suppose our distribution was 8, 10, 8, 14, 40 and 12. When you put the scores in order you get 8, 8, 10, 12, 14, 40 After crossing off the low and high scores, This leaves you with 10 and 12. What would you do?

Rule: When You Have Two Middle Scores, Find Their MEAN 8, 8, 10, 12, 14, 40 Middle Numbers are 10 and 12 Find the Mean: 10 + 12 = 22 22/2 = 11 The Median is 11

What is the mode in the distribution of 8, 10, 8, 14, 40? Definition: The Mode is the score that occurs most frequently in the distribution What is the mode in the distribution of 8, 10, 8, 14, 40?

Frequency Distribution Score Tally Frequency 40 I 1 14 I 1 10 I 1 8 I I 2 Frequency Distribution-a listing of scores from lowest to highest with the number of times each score appears in a sample

Answer to Mode Problem In our distribution of 8, 10, 8, 14, 40, the score 8 appears twice. All other scores appear once Score Tally Frequency 40 I 1 14 I 1 10 I 1 8 I I 2 The Mode is 8

What if Two or More Scores Appear the Same Number of Times? When two scores appear the same number of times, both scores are considered modes When you have two modes, it is a bimodal distribution When you have three or more modes, it is a multimodal distribution When all scores appear the same number of times, there is “No Mode”

Calculate the Mode (s) 1. 8, 10, 8, 10, 14, 40 2. 8, 9, 10, 12, 14, 40, 14, 40, 12, 10, 9, 8

Answer to Both Mode Problems 1. There are two modes-It is a bimodal distribution. The modes are 8 and 10 2. Since all scores appear twice, there is no mode

Calculate the Measures of Central Tendency STUDENT NAME IQ SCORE Billy 105 Juan 125 Carmela   70 Fred 115 Yvonne   85 Amy   105 Carol   95 Sarah 100

Answer to Measures of Central Tendency Question Mean = 100 800/8 = 100 Median = 102.5 100, 125, 70, 115, 85, 105, 95, 100 70, 85, 95, 100, 105, 105, 115, 125, M = 205/2 = 102.5 Median is 102.5 Mode = 105 70, 85, 95, 100, 105, 105, 115, 125,

Range Definition: The Range is the difference between the highest and lowest score in the distribution. To calculate the range, simply take the highest score and subtract the lowest score. In the distribution 8,10,8,14, 40, what is the range?

Answer to Range Problem The Range is 32 High score is 4 Low score is 8 40 – 8 = 32

Problem with the Range The range tells you nothing about the scores in between the high and low scores. Extreme scores can greatly affect the range. e.g., Suppose the distribution was 8, 9, 8, 9, 8, and 1,000. The range would be 992 (1,000 – 8 = 992). Yet, only one score is even close to 992, the 1,000.

Standard Deviation Let’s look at the following two distributions of scores on a 50-question spelling test (each score represents the number of words correctly spelled) Scores for 5 students in Group A: 28, 29, 30, 31, 32 Scores for 5 students in Group B: 10, 20, 30, 40, 50 Calculate the MEAN for Groups A and B

Standard Deviation Mean of Group A = 30 Mean of Group B = 30 The means of both groups are 30. Now, if you knew nothing about these two groups other than their mean scores, you might think they looked similar. However, the spread of scores around the mean in Group A (28 to 32) is much smaller than the spread of scores around the mean Group B (10 to 50).

Standard Deviation There is a statistic that describes for us the spread of scores around the mean Definition: The standard deviation is the spread of scores around the mean. It is an extremely important statistical concept to understand when doing assessment in special education.

Normal Curve A normal distribution hypothetically represents the way test scores would fall if a particular test is given to every single student of the same age or grade in the population for whom the test was designed.

Normal Curve The normal curve (also referred to as the Bell Curve) tells us many important facts about test scores and the population. The beauty of the normal curve is that it never changes. As students, this is great for you because once you memorize it, it will never change on you (and, yes, you do have to memorize it at some point in your academic or professional career).

Percentages Under the Normal Curve 34% of the scores lie between the mean and 1 standard deviation above the mean. An equal proportion of scores (34%) lie between the mean and 1 standard deviation below the mean. Approximately 68% of the scores lie within one standard deviation of the mean (34% + 34% = 68%).

Normal Curve 13.5% of the scores lie between one and two standard deviations above the mean, and between one and two standard deviations below the mean. Approximately 95% of the scores lie within two standard deviations of the mean (13.5% + 34% + 34% + 13.5% = 95%)

The Importance of the Normal Curve Now, how does this help you? Well, let’s take an example that you will come across numerous times in special education: IQ. The mean IQ score on many IQ tests is 100 and the standard deviation is 15

Gifted Programs Do you know what the requirements are for most gifted programs regarding minimum IQ scores (that have a mean of 100 and SD of 15)? By looking at the normal curve you may have figured it out—the minimum is normally an IQ of 130 for entrance. Why? Gifted programs will take only students who are 2 standard deviations or more above the mean. In a sense, they want only those whose IQs are better than 97.5% of the population.

Mental Retardation How about mental retardation? On the Wechsler Scales, the classification of mental retardation is determined if a child receives an IQ score of below 70. Why 70? This score was not just randomly chosen.

Why 70? What we are saying is that in order to receive this classification you normally have to be 2 or more standard deviations below the mean. In a sense, the child’s IQ is only as high as 2.5% (or even lower) of the normal population (or, in other words, 97.5% or more of the population has a higher IQ than this child).

Practice Problem In School district XYZ, the mean score on an exam was 75. The standard deviation was 10. Draw the normal curve for this distribution. Based on the normal curve, what percentage of students scored: between 65 and 85? above 85? between 55 and 95? above 95? 5 and 85

Answer to Practice Problem between 65 and 85? 68% (34 + 34) above 85? 16% (13.5 + 2.5%) between 55 and 95? 95% (13.5% + 34% + 34% + 13.5%) above 95? 2.5%