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

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

Lessons

\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)
or
b) below Q11.5(IQR) Q_1- 1.5(IQR)

  • 1.
    a)
    Z-Score

    b)
    Quartiles

    c)
    InterQuartile Range

    d)
    Percentiles


  • 2.
    Using Z-score to Compare the Variation in Different Populations
    Charlie got a mark of 85 on a math test which had a mean of 75 and a standard deviation of 5. Daisy got a mark of 75 on an English test which had a mean of 69 and a standard deviation of 2. Relative to their respective mean and standard deviation, who got the better grade?

  • 3.
    Determining the Quartiles
    Find the quartiles for each data set:
    a)
    {9, 3, 7, 5, 2, 8, 12}

    b)
    {2, 3, 5, 7, 8, 9, 12, 15}

    c)
    {2, 3, 5, 7, 8, 9, 12, 15, 35}


  • 4.
    Interquartile Range & Box-and-Whisker Plot
    For the data set: {8, 2, 20, 4, 9, 5, 6, 12, 10, 1}
    a)
    Determine the quartiles.

    b)
    Find the interquartile range.

    c)
    Construct a box-and-whisker plot.

    d)
    Which data points, if any, are outliers?


  • 5.
    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}.
    a)
    What percentile did she score in?

    b)
    Sidney’s friend Billy knows he got in the 70% percentile, what was his mark?