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Describing Data Distributions Solutions

• Measures of center help determine single values that can represent    a data set or population   . 

• The measures of center are:

  • Mean (μ): population    average   

  • Median: middle number when the data set is in    ascending order   

  • Mode:    most frequently occurring     number in a data set

• The closer the measures of center are to one another, the more    symmetrically distributed    they are, making the mean the representative measure of center.

•    Technology    should be used to calculate the mean, median, and mode of a data set, to allow for more time to    analyze the data   .

• In statistics, the mean (as well as the variance and standard deviation) is rounded to    one decimal place further    than the raw data.

Note

There are some instances where the mean and median are equal, but the data is still skewed because one or more values of the data set are outliers.

Left-Skewed Distribution

• Generally, the distribution is left-skewed when the $\underline{\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{mean}\mathbf{<}\mathbf{median}\mathbf{ }\mathbf{ }\mathbf{ }}.$ 
• This distribution is also referred to as    negatively    skewed.
• The numbers in the data set are    pulling the mean to the left   , toward negative values.

Right-Skewed Distribution

• Generally, the distribution is right-skewed when the $\underline{ \mathbf{mean}\mathbf{>}\mathbf{median} }$.
• This distribution is also referred to as    positively    skewed.
• The numbers in the data set are    pulling the mean to the right   , toward positive values.

• There are some instances where the mean and median are equal, but the data is still skewed because one or more values of the data set are    outliers   .
• No matter the distribution, it is important to look carefully at both:
  •    the measures of center   
  •    the data set as a whole   
• When distribution is skewed, the median is a better representation of center than the mean because it is    not affected by outliers   .

Note

Outliers are not the focus of this lesson. See Algebra 1: Principles of Secondary Mathematics for more.

Example 1

Name and sketch the distribution.

Example 2

Calculate the measures of center. Then name the distribution.

$\left\{73, 76, 85, 85, 89, 90, 92, 95, 98\right\}$

$\mu :\frac{73+76+85+85+89+90+92+95+98 }{9}=87$

$$\begin{gathered} \mathrm{median}: 89 \mathrm{mean}<\mathrm{median} \\ \mathrm{mode}: 85 \mathrm{left}–\mathrm{skewed} \end{gathered}$$