Standard Scores

Normal Distribution

Normal is used to describe a symmetrical, bell-shaped curve, which has the greatest frequency of scores in the middle, with smaller frequencies towards the extremes

• 50% of the scores occur above the mean, and 50% of the scores occur below the mean

• Approximately 34% of all scores occur between the mean and 1SD above the mean

• Approximately 34% of all scores occur between the mean and 1 SD below the mean

• The area of the normal curve between 2 and 3 standard deviations above the mean is referred to as tail

• The area between -2 and -3 standard deviations below the mean is also referred to as tail

Standard Scores

• raw score that has been converted from one scale to another, the latter scale having some arbitrarily set mean and standard deviation

• standard scores are more readily interpretable than raw scores. With a standard score, the position of a testtaker’s performance relative to other testtakers is readily apparent

• Different systems for standard scores exist: (a) z scores; (b) T scores; (c) IQ; (d) stanine scores

Z Scores

results from the conversion of a raw score into a number indicating how many standard deviation units the raw score is above or below the mean of the distribution

(Knowing that someone obtained a z score of 1 on a spelling test provides context and meaning for the score. Drawing on our knowledge of areas under the normal curve we would know that only about 16% of the testtakers obtained higher scores • By contrast, knowing simply that someone obtained a raw score of 65 on a spelling test conveys virtually no usable information, because information about the context of this score is lacking)

Z Score Formula

z = (X – X με πανω παυλα) / s

X: raw score, Xπαυλα: mean, s: standard deviation

In essence, a z score is equal to the difference between a particular raw score and the mean divided by the standard deviation

Converting a z score to a new standardized score (NSS)

NSS = ASD * (z) + AM

• ASD is the value of the arbitrarily set standard deviation

• z is the value of the standard score to be transformed

• AM is the value of the arbitrarily set mean

T Scores

• T scores correspond to a scale with a mean set at 50 and a standard deviation set at 10

• This standard score system is composed of a scale that ranges from 5 standard deviations below the mean to 5 standard deviations above the mean

IQ Scores

mean set at 100 and a standard deviation set at 15

• The typical mean and standard deviation for IQ tests results in approximately 95% of deviation IQs ranging from 70 to 130. That’s 2 standard deviations below and above the mean respectively

Stanine

• Stanine is a standard score that has a mean of 5 and a standard deviation of approximately 2

• Divided into 9 units, the scale was christened a stanine, deriving from a contraction of the words standard and nine

• Stanines are different from other standard scores in that they take on whole values from 1 to 9, which represent a range of performance that is ½ standard deviation in width

• The 5th stanine indicates performance in the average range, capturing the middle 20% of the scores in the normal distribution

• The 4th and 6th stanines are also ½ standard deviation wide and capture the 17% of cases below and above the 5th stanine respectively

Normality and Standard Scores

Many test developers hope that they will yield a normal distribution of scores in order to be able to produce meaningful and comparable standard scores • Yet even after very large samples have been tested with the instrument under development, skewed distributions might occur

One alternative available is to normalize the distribution = involves “stretching” the skewed curve into a shape of a normal curve and creating a corresponding scale of standard scores • This type of scales are technically referred to as “normalized standard score scales”

The presence of normally distributed scores are desirable for purposes of comparability when using standard scores • One of the primary advantages of a standard score on one test is that it can readily be compared with a standard score on another test