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Measures of relative standing – z-score, quartiles, percentiles - Measures of Dispersion

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Measures of relative standing – z-score, quartiles, percentiles


\cdot zxz_x: z-score, a measure of how many standard deviations a data item xx is from the mean.

population: zx=xμσz_x= \frac{x- \mu}{\sigma}

sample: zx=xxsz_x= \frac{x- \overline{x}}{s}

z-score allows comparison of the variation in different populations/samples.

\cdot Quartiles: values that divide the data set into quarters.

Q1=Q_1= bottom 25% of data
Q2=Q_2= Median == bottom 50% of data
Q3=Q_3= bottom 75% of data

\cdot InterQuartile Range (IQR): represents the middle 50% of the data set.

IQR=Q3Q1IQR= Q_3-Q_1

\cdot Percentiles: indicates what percentage of the data falls below a certain value

PercentileofX=numberofdatapointslessthanXtotalnumberofdatapointsPercentile\;of\;X= \frac{number\;of\;data\;points\;less\;than\;X}{total\;number\;of\;data\;points}

\cdot Outliers: an outlier is a data point which lies an abnormal distance from all other data points.

Outliers are either,

a) above Q3+1.5(IQR) Q_3+1.5(IQR)
b) below Q11.5(IQR) Q_1- 1.5(IQR)

  • 1.
  • 2.
    Determining the Quartiles
    Find the quartiles for each data set:
  • 3.
    Interquartile Range & Box-and-Whisker Plot
    For the data set: {8, 2, 20, 4, 9, 5, 6, 12, 10, 1}
  • 4.
    Determining the Percentile
    Sidney is taking a biology course in university. She got a mark of 78% and the list of all marks from her class (including her mark) is given by {56, 83, 74, 67, 47, 54, 82, 78, 86, 90}.
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Measures of relative standing – z-score, quartiles, percentiles

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