Unit 2 Statistics of One Variable

Transcription

1 Unit 2 Statistics of One Variable Day 6 Summarizing Quantitative Data Summarizing Quantitative Data We have discussed how to display quantitative data in a histogram It is useful to be able to describe how the data in the histogram is distributed (how it looks) Symmetric Distributions U-Shaped Distribution A U-shaped distribution peaks at either end of the range 1

2 Uniform Distribution When each outcome has a similar frequency, it is called a uniform distribution. Mound-Shaped Distribution A mound-shaped distribution is symmetrical about the line passing through the interval with the greatest frequency. Skewed Distributions Left-skewed Right-skewed 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 2

3 Summarizing a Quantitative Distribution 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 1: Summarize the distribution of the following data on the salary structure of Statsville Plush Toys Inc. (Pg 13 #11) Range ($000) Number of Employees Step 1: Identify the variable you are summarizing and give any pertinent information The data describe the salaries of all 100 employees at Statsville Plush Toys Inc. 3

4 Number of Employees 2/20/2014 Step 2: Make a histogram and a box-and-whisker plot 3 Distribution of Employees at Statsville Plush Toys Inc in Thousands ($) For the box and whisker plot, we will need to locate the quartiles of the data Range ($000) Number of Employees median = 4 between 0th and 1 st values Q1 = 3 between 2 th and 26 th values Q3 = between 7 th and 76 th values There are 100 employees 2 th and 26 th values are in this bin 0 th and 1 st values are in this bin 7 th and 76 th values are in this bin 4

5 Number of Employees 2/20/ Distribution of Employees at Statsville Plush Toys Inc in Thousands ($) Step 3: Based on the distribution of the data, choose appropriate numerical summaries What Number Should I Use? Center 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 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)

6 Number of Employees 2/20/ Distribution of Employees at Statsville Plush Toys Inc. median = $ in Thousands ($) IQR = Q3 Q1 = = $ The data is skewed so we will report the median and the IQR but check the mean and standard deviation as well to see the effect of the outlier Mean and standard deviation must be done using the grouped data formulas Range ($000) Midpoint m Number of Employees f m f (m μ) f m μ μ = Total 100 Total mf N = = 4.3 σ = f m μ 2 N = =

7 Step 4: Summarize and interpret your findings. Discuss the shape of the distribution, center, spread and any unusual features The data is unimodal and somewhat right-skewed with a median of $ and and IQR of $ indicating that the spread of employee salaries at Statsville Plush Toys Inc. is fairly high There was a high outlier of $ In this case the outlier did not have a significant effect on the mean of $ From the standard deviation it can be concluded that on average, employee salaries differ from the mean by $ Summarizing Quantitative Data Handout Take a look at a reference guide on avoiding common mistakes when analysing data ask your teacher where to find it! Test Tuesday 7