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 give Always 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 roughly 68% 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 SkewedRight 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.lower quartile 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.upper quartile 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

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