Introduction to Statistics for Built Environment

Course Code: AED 1222Compiled by

DEPARTMENT OF ARCHITECTURE AND ENVIRONMENTAL DESIGN (AED)CENTRE FOR FOUNDATION STUDIES (CFS)

INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA

Lecture 10

Normal Curve

Today’s Lecture: Calculating the z-score Interpreting the z-score Areas under the normal curve Skewness

Z-Scores – a Measure of Position

• The z -score is also called the standard score.

• It is an ordinary score converted so that it better describes that score's place in its distribution.

• It denotes the number of standard deviations a data value is above the mean (if +) or below the mean (if -).

• Why z-score?– As a scale.– Provides a generalized standard of comparison.

Calculating the z-score

Z-score formula for sample data

Z-score formula for population data

Z =s

x - x

Z = x - µs

This value is a measure of the distance in standard deviations of a data value from the mean.

5

Calculating Z – Scores: an example

Mansur is interested to become a salesperson to gain basic marketing skills.

During an interview, he took a sales skill test and scored 135.Does this indicate he has the potential for sales?

If average score of this test is 100, his score (135) is above average. But how much is above average? Slightly, moderately or dramatically?

How high is Mansur’s score as compared to anyone else who took the same test?

Given standard deviation is 25, convert Mansur’s score to a z-score.

Z = 1.4

Original score New score/Z-Score

Z =x - µ

s Z = 135 - 100

25

Calculating z-scores cont.

7

The same case, continued

At the interview, Mansur noticed there’s another vacant position in research analysis.

He thought why not having a try in case he does not make it in the first interview. In a test to apply for this post, he scored 85.

The average score (mean) of this test is 60 and the standard deviation is 10.

Calculating z-scores cont.

Let’s see Mansur’s z-score in the second test>>

Z = x - µs

Z = 85 - 6010

Z = 2.5

Original score New score/Z-Score

So Mansur scored 2.5 std. deviations above the mean in research analysis test but he scored only 1.4 std. deviations above the mean in sales skill test.It seems that Mansur’s inclinations is more towards research analysis.

Calculating z-scores cont.

Interpreting z-scores

z-score < 0Is when a data value is less than the mean

z -score > 0Is when a data value is greater than the mean

z -score = 0Is when a data value is equal to the mean

A normal distribution is a bell-shaped distribution of data where the mean, median and mode all coincide. Chebyshev’s theorem applies to any distribution regardless of its shape. However, when a distribution is bell-shaped, the following statements which make up the empirical rule, are true.

Approximately:

•68% of the data values will fall within 1 sd. Of the mean.

•95% of the data values will fall within 2 sd. Of the mean.

•99.7% of the data values will fall within 3 sd. Of the mean.

For example:

The score on a national achievement exam have a mean of 480 and a deviation of 90. If these scores are normally distributed, then approximately 68% will fall between 390 and 570 (480+90=570 and 480-90=390)

480390 570

Skewness

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.

Mean Mode

Positively skewed/ skewed to the right

If there are extreme values towards the positive end of a distribution, the distribution is said to be positively skewed. In a positively skewed distribution, the mean is greater than the mode. For example:

Negatively skewed/ skewed to the left

A negatively skewed distribution, on the other hand, has a mean which is less than the mode because of the presence of extreme values at the negative end of the distribution.

Calculating skewness

There are a number of ways of measuring skewness:

Pearson’s coefficient of skewness = mean – mode = 3(mean – median)

Standard deviation Standard deviation

The value determines whether the data distribution is positively skewed or negatively skewed. If the value is positive, it means the distribution is positively/rightly skewed. If the value is negative, it means the distribution is negatively/ leftly skewed.