Summarizing Quantitative Data

Summarizing Quantitative Data

• We have discussed how to display data in a histogram.

• Today learn to describe how data is distributed.

Symmetric Distributions

• U –Shaped Distributions or Bimodal Distribution

Peaks at either end of the range.

Bimodal Distributions

Bimodal Distributions

Bimodal Distribution

Uniform Distribution

• When each outcome has a similar frequency it is called a uniform distribution.

• Most common occurrence is in situations with total randomness.

Uniform Distribution

Normal Distribution

• Mound shape or “Bell Curve”• Distribution found in MANY applications in Science, commerce,

industry, etc.• Blood Pressure, heart rates, height of adult males, weights of 2 year

old girls, test scores, etc.

Normal Distribution

Skewed Distributions

• In a skewed distribution, the interval or group of intervals that contains the greatest frequencies is near one end of the histogram • The thinner ends of the distribution are called the tails • If one tail stretches out farther than the other, the histogram is said

to be skewed to the side of the longer tail

Summarizing Data

• Step 1: Identify the variable you are summarizing and give any pertinent information • Step 2: Make a histogram and a box-and-whisker plot • Step 3: Based on the distribution of the data, choose appropriate

numerical summaries • Step 4: Summarize and interpret your findings. Discuss the shape of

the distribution, center, spread and any unusual features (eg. Outliers)

Example

• Summarize the distribution of the following data on the salary structure of Statsville Plush Toys Inc.

Step 1 Identify Variable

• Step 1: Identify the variable you are summarizing and give any pertinent information

The Data Describes the salaries of all 100

employees at Statsville toy companies

Step 2 Make Histogram; Box and Whisker

Median: 50th and 51 entry are here3rd Quartile: 75th and

76th entry are here

1st Quartile: 25th and 26th entry are here

Step 2 Make Histogram; Box and Whisker

Step 3 Pick Appropriate Numerical Summaries• Centre and Spread• Always pair the median with the IQR and the mean with the standard

deviation • If the distribution is skewed, report the median and the IQR. You may wish to

point out the mean and standard deviation as well, but point out why the mean and median differ • If the distribution is symmetric, report the mean and standard deviation

Step 3 Pick Appropriate Numerical Summaries• Unusual Features• If there are multiple modes, try to identify why • If there are clear outliers, point them out. If you are reporting mean and

standard deviation, report them with outliers present and with outliers removed (median and IQR not affected much by outliers)

Step 4 Summarize and interpret

• The data is unimodal and somewhat right-skewed with a median of $ 45 000 and IQR of $ 20 000 indicating that the spread of employee salaries at Statsville Plush Toys Inc. is fairly high • There was a high outlier of $ 95 000. In this case the outlier did not

have a significant effect on the mean of $ 45 300 • From the standard deviation it can be concluded that on average,

employee salaries differ from the mean by $ 13 380

Assignment

• Handout