$\cdot$ **$z_x$: 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

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:

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?