Dec 6 B1 BW Standard deviation and z-scores

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how much the values in a dataset deviate from the mean.

The formula for standard deviation (σ) is: $$\text{σ} = \frac{\text{√Σ(x - μ)²}}{N}$$ , where μ is the mean, x represents each value, and N is the number of values.

A z-score indicates how many standard deviations a data point is from the mean. It helps to understand the position of a value within a distribution.

The formula for calculating a z-score is: $$z = \frac{x - μ}{σ}$$ , where x is the value, μ is the mean, and σ is the standard deviation.

$$z = \frac{50 - 40}{5} = 2$$ .

A z-score of 0 indicates that the data point is exactly at the mean of the distribution.

A positive z-score indicates that the data point is above the mean.

A negative z-score indicates that the data point is below the mean.

This implies that the student's score is 1.5 standard deviations above the mean score of the test.

In a normal distribution, z-scores can be used to determine the probability of a score occurring within that distribution.