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

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

Lessons

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

population: $z_x= \frac{x- \mu}{\sigma}$

sample: $z_x= \frac{x- \overline{x}}{s}$

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

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

$Q_1=$ bottom 25% of data
$Q_2=$ Median $=$ bottom 50% of data
$Q_3=$ bottom 75% of data

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

$IQR= Q_3-Q_1$

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

$Percentile\;of\;X= \frac{number\;of\;data\;points\;less\;than\;X}{total\;number\;of\;data\;points}$

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

Outliers are either,

a) above $Q_3+1.5(IQR)$
or
b) below $Q_1- 1.5(IQR)$

• Introduction
a)
Z-Score

b)
Quartiles

c)
InterQuartile Range

d)
Percentiles

• 1.
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?

• 2.
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}

• 3.
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?

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

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