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148 CHAPTER 2 • Modeling One-Variable Quantitative Data

CHAPTER 2 LEARNING
TARGETS GRID Chapter 2

You can nd a grid with all of the
learning targets for this chapter by Main Points
clicking on the link in the TE-book or by
logging into the teachers’ resources on
our digital platform. An extra column Describing Location in a Distribution measures of center, location, and variability are

has been added for students to track ■ Two ways of describing an individual data value’s multiplied (divided) by b .

their progress. The learning targets grid location in a distribution of quantitative data are ■ Neither of these transformations changes the

shape of the distribution.
percentiles and standardized scores ( -scores).
z

is a great way to help students take ■ An individual’s percentile is the percent of values ■ A common transformation is to standardize

ownership of their learning. in a distribution that are less than the individual’s all the values in a distribution: For each value,
(C) 2021 BFW Publishers -- for review purposes only.
data value. subtract the mean of the distribution and then
divide the difference by the standard deviation.
■ To standardize any data value, subtract the mean
of the distribution and then divide the difference The transformed distribution has the same shape
by the standard deviation. The resulting standard- as the original distribution, a mean of 0, and a
ized score ( z -score) standard deviation of 1.
value − mean
z = Density Curves
standard deviation

■ We can describe the overall pattern of some distri-
measures how many standard deviations the butions of quantitative data with a density curve .
data value lies above or below the mean of the A density curve is an idealized description of the
distribution. distribution that smooths out the irregularities in

■ We can also use percentiles and z -scores to com- the actual data. A density curve always remains
pare the location of individuals in different distri- on or above the horizontal axis and has total area
butions of quantitative data. 1 underneath it. An area under a density curve

■ In the special case of a normal distribution, we and above any interval of values on the horizontal
can convert from percentiles to z -scores and from axis estimates the proportion of all observations
z -scores to percentiles. that fall in that interval.

■ We write the mean of a density curve as µ and the

Transforming Data standard deviation of a density curve as σ to dis-

tinguish them from the mean x and the standard

It is necessary to transform data when changing units deviation s x of the sample data.

of measurement. ■ The mean and the median of a density curve can

■ When you add a positive constant a to (subtract be located by eye. The mean µ is the balance point

a from) all the values in a quantitative data set, of the curve. The median divides the area under

measures of center and location—mean, median, the curve in half. The standard deviation σ can-

percentiles—increase (decrease) by a Measures of not be located by eye on most density curves.
.
variability—range, standard deviation, IQR —do ■ The mean and median are equal for symmetric

not change. density curves. The mean of a skewed density
■ When you multiply (divide) all the values in a curve is located farther toward the long tail than

quantitative data set by a positive constant b , the median is.
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148 CHAPTER 2 • Modeling One-Variable Quantitative Data

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