CHAPTER EIGHT

MEASURES OF

RELATIVE POSIT

ION

8.1Intro

duction

the previous

chapter, you

learnt how to

compute the variance and standard deviation. This chapter will expose you to Measures of Relative Position. This indicates where a score is in the distribution. It permits one to compare the performance of an individual with all other individuals in the same sample, all measured on the same variable. Measures of Relative Position that will be discussed here are Z-scores and T-scores.

8.2Objectives

At the end of this chapter, you should be able to:

define Z-scores
define T-scores
compute Z-scores and T-scores.
List and explain the properties of Z-score and T-score.
To represent skewedness of a distribution
Represent kurtosis of a distribution

8.3Z-Score

It is a straightforward standard score that translates test performance into a raw score above or below the mean, or the amount of standard deviation units. The formula below is used to determine the Z-score:

Z–score =\\frac{\\mathbf{X}-\\overline{\\mathbf{X}}}{\\mathbf{S}\\mathbf{D}}

Where X = raw score

\\overline{\\mathbf{X}} = mean of the raw scores

SD = standard deviation

If this is not taken into account in the test interpretation, the fact that the Z-score is negative because the raw score is less than the mean might have major consequences. As a result, Z-scores are transformed into a common scoring scheme that only accepts non-zero and positive values. Z-scores often fall within a range– 4 to + 4 (–4< Z< 4).

8.4T-Score

T–score is linear transformation of Z–score into a higher number or index using the formula. T–score = 10Z + 50

8.5Computation of Z–score and T–score

A test with mean score 40 and a standard deviation was 4. Calculate T–scores of two tests with raw scores of 45 and 30 respectively in the test.

Solution

To compute for T-scores, you must firstly calculate Z-scores for the testee. Then the Z-scores would be converted to the T–score required.

The raw score of 45, the Z—score is

Z–score = \\frac{\\mathbf{X}-\\overline{\\mathbf{X}}}{\\mathbf{S}\\mathbf{D}} where X = 45, \\overline{\\mathbf{X}} = 40, SD = 4

Z–score = \\frac{45\\hbox{--}40}{4} =\\frac{5}{4}=1.25 then

T–score = 50 + 10(Z),Z—score = 1.25

= 50 + 10 (1.25)

= 50 + 12.5

= 62.5

The raw score of 30, the Z–score is

Z–score = \\frac{\\mathrm{X}-\\overline{\\mathrm{X}}}{\\mathrm{S}\\mathrm{D}} where X = 30, \\overline{\\mathrm{X}} = 40, SD = 4

=\\frac{30\\hbox{--}40}{4} = \\frac{\\hbox{--}10}{4}

= \\hbox{--}2.5

Now the T–score = 50 + 10 (Z)Z—score = \\hbox{--}2.5

= 50 + 10 (- 2.5)

= 50—25

= 25

8.5.1Properties of Z-score and T-score

The following are properties (characteristics) of the two scores.

Z-Score
Range; generally, as 4.0\\leq Z \\leq 4.0
Mean; \\overline{z}=0 and lastly,
Standard deviation as S.\\mathrm{D}_{Z}=1.00
T-Score
Range; generally, as 10\\leq T\\leq 90
Mean; T=50 and lastly,
Standard deviation as S.\\mathrm{D}_{T}=10

8.6Normal Distribution

Normal Distribution is sometimes known as Gaussian distribution and is a continuous probability distribution, where in values under the curve is always 1 or 100%.

The formula for the normal probability density function seen complicated. You only need to know the population mean and standard deviation when to use it.

If it is represented in graph form, the normal distribution appears as a bell curve. A normal distribution is symmetrical, but all symmetrical distribution is normal. Many naturally occurring phenomenon tend to approximate the normal distribution but in finance, most pricing distributions are not however, perfectly normal.

The formula for a normal distribution is given as:

f\\left(x\\right)=\\frac{1}{\\delta \\sqrt[]{2\\overline{\\wedge }}}e^{\\sfrac{\\hbox{--}1}{2}}\\frac{(x-\\mathrm{\\mu})^{2}}{\\delta }

Where x= value of the variable & f\\left(x\\right)= the probability function

\\mathrm{\\mu}= The mean

\\delta = The standard deviation.

\\overline{\\wedge }=3.142 Approximate.

e=2.718 Approximate.

The Normal Distribution curve is a frequency polygon and symmetrical about a point at which the mean, median and mode are all equal. That is, it is symmetrical about a maximum.

Many variables form a normal distribution, including physical measures, such as height, distance, weight, and psychological measures, such as intelligence and aptitude. Most variables measurement in education from normal distribution if enough subject is tested.

8.6.1Skewed Distribution

A distribution that is not normal, non-symmetric, then the value of mean, median and mode are not equal and that it is said to skewed distribution Skewedness is the degree of a symmetric and there are of two type’s skewedness-positive skewedness and negative skewedness.

Positive Skewedness

A distribution skewed to the right that is when the longer tail for the curve occur to the right. The mean is greater than the median which in turn greater than the mode.

mean

median

mode

mean

median

mode

Fig 6.1 Positive Skewed Distribution

In a positive skewed distribution, the mean has the highest value, then the median and followed by mode.

This distribution take place when test is too difficult or not properly rated.

Negative Skewedness:

Negative skewedness take place if the longer tail of the curve falls to the left. That is the mean value is of the lowest than followed by the median and mode has the highest value. This can occur where there is too simple test.

mode

median

mean

mode

median

mean

Fig 6.2 Negative Skewed Distribution

The formula for coefficient of skewedness is as:

S=\\frac{\\sum \\left(x-\\overline{x})^{2}}{\\left(\\frac{\\sum \\left(x-\\overline{x})^{2}}{N})^{3}}

A normal distribution, s=0

Positive skewedness =\\left(+\\right)

Negative skewedness =\\left(-\\right)

8.6.2Kurtosis

Kurtosis is a work derived from Greek word meaning '‘curved’‘.

It is the degree of Peakedness of a distribution or a measure that indicates the concentration of the score close to the mean. Measurement of the degree of Peakedness of a Distribution is known as coefficient of kurtosis. Computation for coefficient of kurtosis is given as

k=\\frac{\\sum \\left(x-\\overline{x})^{4}}{\\left(\\sqrt[]{\\frac{\\sum \\left(x-\\overline{x})^{2}}{N}})^{4}}

00When the coefficient of kurtosis IS K = 3. It is Mesokurtic. When the coefficient of kurtosis is greater than 3(k>3), the distribution is Leptokurtic, while when it is less than 3 (k<3), it is Platykurtic

Fig 6.3a Fig 6.3bFig 6.3c

Fig 6.3a—Leptokurtic

Fig 6.3b—Platykurtic

Fig 6.3c—Mesokurtic

Student Activity

A distribution of percentage scores has a mean 50 and a standard deviation of 15. If two percentage scores from the distribution were converted to T–scores as:
20 and (ii) 80, find the raw scores.

A Test’s standarda deviation is 5, with a mean score of 50. Determine te T-Scores for two tastes who performed with raw scores of 25 and 10 in the test, respectively.

Scores	Frequency
9—28	9
19—28	1
29—38	1
39—48	10
49—58	25
59—68	25
69—78	28
79—88	15
89—98	10
99—108	1
109—118	1

3. The follwing is the distrobitopm pf the scpres pf 12- SSII students in posttest.

In this distribution if you discover that the mean, mode and median are all equal comment.
If this distribution is leptokurtic, platykurtic or Mesokurtic. Justify in a single line to explain the reason for your choice.
Consider the values of this distribution in the table below

Mean	Median	Mode
70	65	60
14	14	14
50	62	70

In each case how would you describe the distribution of the observation, using the terms symmetric, positively skewed negatively skewed.

References

Maruf, O.I & Aliyu, Z. (2013). Measurement and Evaluation in Education. Printed by Stevano Printing Press, General Printers, and Publisher.

National Teachers’ institute & National open University of Nigeria (2016). General Education Course.