The 68 95 99 7 rule and z scores
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The 68-95-99.7% Rule and z-scores. Normal distribution. The shape of histograms sometimes approximate what is known as a “bell-shape.”. Normal distribution. By joining the columns of the histogram together, a bell-shaped curve is formed.

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Normal distribution
Normal distribution

  • The shape of histograms sometimes approximate what is known as a “bell-shape.”


Normal distribution1
Normal distribution

  • By joining the columns of the histogram together, a bell-shaped curve is formed.

  • When we get data that forms a bell-shape, we call it a normal distribution.

  • Two important things to remember about normal distributions:

    • The mean of the data, , is at the centre of the bell-curve.

    • Most of the data is gathered close to the mean. The further we go from the mean the less data there is.


The 68 95 99 7 rule
The 68-95-99.7% rule

  • Recall that standard deviation, s, is a measure of the spread of the data.

  • In a normal distribution,

    68% of data have values within 1 standard deviation of the mean.


The 68 95 99 7 rule1
The 68-95-99.7% rule

95% of the data have values within 2 standard deviations of the mean.


The 68 95 99 7 rule2
The 68-95-99.7% rule

99.7% of the data have values within 3 standard deviations of the mean.


The 68 95 99 7 rule3
The 68-95-99.7% rule

  • Using the 68-95-99.7% rule, we can work out the percentage of data in each section of the bell curve.


Questions
Questions

  • Examples on board 1I, Q. 2, 3, 4, 10, 13

  • Questions: 5, 7, 9, 11, 14


Z scores
z-scores

  • The z-score indicates the position of a certain score in a set of data relative to the mean.

  • It enables us to compare scores in different distributions.

  • A z-score of zero means that the score is equal to the mean. A negative z-score means that the score is below the mean, and a positive z-score is a score above the mean.


Z scores1
z-scores

  • Calculate z-scores using the formula

    where x is the score, is the mean, and s is the standard deviation.

  • A z-score of -1 would mean that the score is one standard deviation below the mean. A score that is one standard deviation above the mean would have a z-score of +1.


Questions1
Questions

  • Examples: 15, 20, 23

  • Questions: 16, 19, 22, 25


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