2.1 Visualizing Distributions:Shape, Center, and Spread

The student will be able:

• To identify and sketch the basic shapes of distributions of data – uniform, normal, skewed

• To describe the characteristics of the shape of a distribution, including symmetry, skewness, modes, outliers, gaps, and clusters.

• To describe a uniform distribution using the range and the frequency.

• To estimate graphically the mean and standard deviation of a normal distribution and use them to describe the distribution.

• The estimate graphically the median and quartiles and use them to describe a skewed distribution.

Important Terms and Concepts• Basic shape of a distribution (listed on next slide)• Measure of center– Mean– Median

• Measures of spread– Standard deviation– Quartiles

• Other features– Outliers– Gaps– Clusters

Important Terms and Concepts

Shape, Center, and Spread – Always giveAlways label graphs!

Four Most Common Shapes of Distribution

• Uniform (Rectangular) Distributions• Normal Distributions• Skewed Distributions• Bimodal Distributions

Uniform Distributions

• All values occur equally often or nearly equally often

Normal Distribution

Bell shaped Single peak

Mode At line of

symmetry

Normal Distribution

• The curve drops off smoothly on both sides, flattening towards the x-axis but never quite reaching it and stretching infinitely far in both directions

Normal Distribution

• On either side of the mode are inflection points – where the curve changes from concave down to concave up

Normal Distribution

• You should use the mean to describe the center• You should use the Standard Deviation, SD, to describe

spread.• The Standard Deviation – the horizontal distance from

the mean to an inflection point.• Use area to estimate the standard deviation.• Roughly 68% of the total area under the curve is

between the vertical lines through the two inflection points. – In other words, the interval between one standard deviation

on either side of the mean accounts for roughly68% of the area under the normal curve

Normal Distribution

• Discuss measure the diameter of a tennis ball• Discuss Page 31, weight of pennies

Graph a Normal Distribution

Skewed Distributions

• Distribution with bunching a one end and a long tail stretching out in the other direction.

• The direction of the tail tells whether the distribution is skewed right or skewed left.

Left Skewed Right Skewed

Skewed Distributions

Skewed

• Since there is no symmetry the ideas of center and spread are not as clear-cut as they are for a normal distribution.

• Typically you should use median to describe center.

• You should use the lower and upper quartile to indicate spread.

What is Lower and Upper Quartile?

• The lower quartile is the value that divides the lower half of the distribution into two halves, with equal numbers of dots on either side.

• The upper quartile is the value that divides the upper half of the distribution into two halves, with equal numbers of dots on either side.

• The three values—lower quartile, median, and upper quartile—divide the distribution into quarters. This allows you to describe a distribution as in the introduction to this chapter: “The middle 50% of the SAT math scores were between 630 and 720, with half above 680 and half below.”

Skewed Distributions

Bimodal Distributions

• Bimodal Distributions have two or more obvious peaks.

• It is worth asking whether your cases represent two or more groups.

Bimodal Distributions

Bimodal Distributions

Other Features

• Outliers – a value that stand apart from the bulk of the data. An unusual value.

• Gaps – a separation – there is no formal definition

• Clusters – a grouping of values – there is no formal definition.

Other Features

Practice

• P1 – P5

Entertainment

• E1 – E8, E11, E14