Sunday, April 9, 20231

Posttest day 3Posttest day 31. Central Tendency ....

2. Mode is.....

3. Median is...

4. Mean is....

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Is used to talk about the central point in the distribution of values in the data.

The measure of central tendency which reports the most frequently obtained score in the data.

The score which is at the center of the distribution. Half the scores are above the median and half are below it.

The arithmetic average of all scores in a data set.

Measures of Central TendencyMeasures of Central Tendency

• Mean: arithmetic average of all scores in a distribution

• Median: the point at which exactly half of the scores in a distribution are below & half are above

• Mode: most frequently occurring score(s)

Normal CurveNormal Curve

Objectives

1.Introduce the Normal Distribution

2.Properties of the Standard Normal Distribution

3.The bell-shaped distribution

4.Z-scores

5.T-scores

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Normal DistributionNormal Distribution

For years, scientists have noted that many variables in the behavioural and physical sciences are distributed in a bell shape.

These variables are normally distributed in the population, and their graphic representation is referred to as the normal curve or a bell-shaped distribution

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Normal DistributionNormal Distribution

If there is no very extreme scores and if you have 30 or more observations, you may have a normal distribution.

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The beauty of the normal curve: The beauty of the normal curve:

No matter what Mean and SD are, the area between Mean - SD and Mean + SD is about 68%; the area between Mean - 2SD and Mean+2SD is about 95%; and the area between Mean - 3SD and Mean + 3SD is about 99.7%. Almost all values fall within 3 standard deviations.

68-95-99.7 Rule68-95-99.7 Rule

68% of the data

95% of the data

99.7% of the data

Normal /bell-shaped curve

Properties of Normal /bell-shaped curve

• It is a symmetrical distribution• Most of the scores tend to occur near the center

– while more extreme scores on either side of the center become increasingly rare.

– As the distance from the center increases, the frequency of scores decreases.

• The mean, median, and mode are the same.

The normal distributionThe normal distributionThe normal distribution is actually a group of

distribution, each determined by a mean and a standard deviation.

A normal distribution means that most of the scores cluster around the midpoint of the distribution, and the number of scores gradually decrease on either side of the midpoint.

The resulting polygon is a bell-shaped curve

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The normal distributionThe normal distributionWhy are normal distributions so

important?Many dependent variables are

commonly assumed to be normally distributed in the population

If a variable is approximately normally distributed we can make inferences about values of that variable

Example: Sampling distribution of the mean

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Key Areas under the CurveKey Areas under the Curve

For normal distributions+ 1 SD ~ 68%+ 2 SD ~ 95%+ 3 SD ~ 99.9%

Normal /bell-shaped curve

Negative Skew Positive skew

Test items were easy.

Testees performed well.

The score are far from zero.

Test items were difficult.

Testees performed poorly.

The scores are near zero.

Skewed DistributionSkewed Distribution

The 68/95/99.7 Rule The 68/95/99.7 Rule

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The normal distributionThe normal distributionRegardless of the exact shapes of the normaldistributions, all share four characteristics:1. The curve is symmetrical around the vertical axis

(half the scores are on the right side of the axis, and half the scores are on its left).

2. The scores tend to cluster around the center (i.e., around the mean, or the vertical axis in the center).

3. The mode, median, and mean have the same values.

4. The curve has no boundaries on either side (the tails of the distribution are getting very close to the horizontal axis, but never quite touch it).*

25 Keep in mind that this is a theoretical model. In reality, the number of scores in a given distribution is finite, and certain scores are the highest and the lowest points of that distribution

Standard ScoresStandard Scores

Two types of scores:

1.Individual scores (raw scores): scores are obtained by individuals on a certain measure.

2.Group scores (mode, median, mean, range, variance, and standard deviation): are summary scores that are obtained for a group of scores.However, both types of scores are scale specific and cannot be used to compared scores on two different measures, each with its own mean and standard deviation

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To illustrate this point, let’s look at the following example.

To illustrate this point, let’s look at the following example.

Suppose we want to compare the scores obtained by a student on two achievement tests, one in English and one in mathematics, Let’s say that the student received a score of 50 in English and 68 in mathematics. Because the two tests are different, we cannot conclude that the student performed better in mathematics than In English. Knowing the student’s score on each test will not allow you to determine on which test the student performed better. We do not know, for example, how many items were on each test, how difficult the test were, and how well the other students did on the tests. Simply put, the two tests are not comparable.

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To illustrate this point, let’s look at the following example.

To illustrate this point, let’s look at the following example.

To be able to compare scores from different tests, we can first convert them into standard scores.

A standard scores is a derived scale score that expresses the distance of the original score from the mean in standard deviation units.

Once the scores are measured using the same units, they can then be compared to each other.

Two types of standard scores are discussed in this slides : z scores and T scores. (T scores are not related to the t-test that will be discussed later).

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Z-ScoresZ-Scores

The z score is a type of standard score that indicates how many standard deviation units a given score is above or below the mean for that group.

The z scores create a scale with a mean of 0 and a standard deviation of 1.

The shape of the z score distribution is the same as that of the raw scores used to calculate the z scores.

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Standard ScoresTo compare scores on different

measurement scales

Z-Scores: the commonest score

Z-score propertiesHow many scores above/below the meanThe mean being set at zeroThe SD being set at one

Z-ScoresZ-Scores

To convert a raw score to a z score, the raw score as well as the group mean and standard deviation are used. The conversion formula is:

Where X= Raw score

= Group mean

SD= Group standard deviation

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Table 1 below presents the raw scores of one student on four tests (social studies, language arts, mathematics, and reading). The table also displays the means and standard deviations of the student’s classmates on these test and shows the process for converting raw scores into z scores.

Table 1 below presents the raw scores of one student on four tests (social studies, language arts, mathematics, and reading). The table also displays the means and standard deviations of the student’s classmates on these test and shows the process for converting raw scores into z scores.

Subject Raw Score

Mean SD Z score

Social studies 85 70 14

Language arts 57 63 12

Mathematics 65 72 16

Reading 80 50 15

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Table 1 Student’s score, Class Means, Class standard Deviations, and z Scores on Four Tests

T-scoreT-score The T score is another standard score measured

on a scale with a mean of 50 and a SD of 10. In order to calculate T scores, z scores have to be

calculate first. Using this standard score overcomes problems

associated with z scores. All the scores on the T score are positive and range from 10 to 90.

Additionally, they can be reported in whole numbers instead of decimal points.

In order to convert scores from z to T, we multiply each z score by 10 and add a constant of 50 to that product. This is the formula:

Standard Scores

T-Score: A standard score whose distribution has a mean of 50 and a standard deviation of 10.

Advantages of T-scoreEnabling us to work with whole numbersAvoiding describing subjects’

performances with negative numbers

Standard Scores

T-Score: A standard score whose distribution has a mean of 50 and a standard deviation of 10.

Advantages of T-scoreEnabling us to work with whole numbersAvoiding describing subjects’

performances with negative numbers

Table 2 conversion of z Scores to T ScoresTable 2 conversion of z Scores to T Scores

Subject Z score T score

Social studies +1.07 10(+1.07) + 50= 60.7 or 61

Language arts -0.50 10 (-0.5) + 50 = 45.0 or 45

Mathematics -0.44 10 (-0.44) + 50 = 45.6 or 46

Reading +2.00 10 (+2.00) + 50 = 70.0 or 70

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References References Main Sources

Coolidge, F. L.2000. Statistics: A gentle introduction. London: Sage.Kranzler, G & Moursund, J .1999. Statistics for the terrified. (2nd ed.). Upper Saddle River, NJ: Prentice Hall.Butler Christopher.1985. Statistics in Linguistics. Oxford: Basil Blackwell.Hatch Evelyn & Hossein Farhady.1982. Research design and Statistics for Applied Linguistics. Massachusetts: Newbury House Publishers, Inc.Ravid Ruth.2011. Practical Statistics for Educators, fourth Ed. New York: Rowman & Littlefield Publisher, Inc.Quirk Thomas. 2012. Excel 2010 for Educational and Psychological Statistics: A Guide to Solving Practical Problem. New York: Springer.

Other relevant sources

Field, A. (2005). Discovering statistics using SPSS (2nd ed.). London: Sage. Moore, D. S. (2000). The basic practice of statistics (2nd ed.). New York: W. H. Freeman and Company.   

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