STANDERDISED VARIABLES ( Z SCORE) FOR STATISTICS GCE O LEVELS/IGCSE

A variable is defined to be Standardized or in standard units if it is expressed in term of deviations from its mean and divided by its standard deviation. It is denoted by Z.

Z=(x-µ)/ δ

The Z-values, being independent of the units of measurement, provided a basis for comparison between individual values, even though they belong to different distributions. That is why the often used in psychological and educational testing, where they are known as standard scores.

Standard Score The standard score is a measure that locates the position of a particular observation with reference to the mean and the standard deviation.

Let x and y be the marks on the original scale and the new scale respectively.

(x-µx)/ δx=(y- µy)/ δy

Where

x is original/unscaled/actual /real value,

y is new/standardised/scaled value,

µx is the actual mean value

µy is the new/Standardized/scaled mean value,

δx is the actual standard deviation,

δy is the new/Standardized/scaled standard deviation.

PROPERTIES OF Z,-SCORE

1. Z-scores are free of units.
2. The mean of Z-score is always zero.
3. The standard deviation of Z-score is always 1.
4. The distribution of Z.-score looks exactly the same as the distribution of the original data.