Chapter 4 Displaying & Summarizing Quantitative Data

Chapter 4

Displaying & Summarizing Quantitative Data

Histograms

Similar to bar charts, but with quantitative data.

No gaps between bars.

Summarizes data visually using frequency count.

Data: Amount spent by 50 customers at a grocery store

2.32 6.61 6.90 8.04 9.45 10.26 11.34

11.63 12.66 12.95 13.67 13.72 14.35 14.52

14.55 15.01 15.33 16.55 17.15 18.22 18.30

18.71 19.54 19.55 20.58 20.89 20.91 21.1323.85 26.04 27.07 28.76 29.15 30.54 31.9932.82 33.26 33.80 34.76 36.22 37.52 39.2840.80 43.97 45.58 52.36 61.57 63.85 64.3069.49

Source:

http://lib.stat.cmu.edu/DASL/Datafiles/Shoppers.html

Histogram: Grocery Data

Histogram: Heights of Adolescents

Source:

http://wiki.stat.ucla.edu/socr/index.php/SOCR_Data_Dinov_020108_HeightsWeights

Histogram: Smiling Times of 8-week old baby

Data Source

:

http://cnx.org/content/m16819/latest/

Stem-and-Leaf Display

Quick way to summarize a small set of quantitative data.

99, 53, 93 , 82 , 85 , 64 , 75 , 62 , 74 , 81 , 73 , 70 , 81 , 73 , 94, 67 , 93 , 87 , 85 , 36 , 80 , 78

Shape of a Distribution

Unimodal

One peak value that occurs more frequently than the rest

Bimodal

Two peak values that occur more frequently than the rest

MultimodalThree or more peak valuesUniformBars in histogram are all about the same height

Symmetry

Does the data look symmetric relative to the middle?

Does the distribution of the left half look like the right half?

Is the data skewed?

Are there tails on the data that stretch out away from the center?

Skewed to the Left: tail is on the leftSkewed to the Right: tail is on the right

Any unusual features (outliers)?

Sometimes a small number of data values are significantly far away from the rest.

Sometimes they can be a mistake in the data but can also be legitimate values that can be left out with a good explanation.

Center of the Distribution: Median

Once we’ve described the basic shape, we want to be able to talk about the center.

Use the horizontal axis to try to identify the center or median.

Half the data above the median, half below

Spread of the Data

Range: max – min

Only takes into account the very extremes, doesn’t measure spread in between

Interquartile

Range (IQR)

Quartiles divide data into quartiles (quarters)Lower Quartile: separates bottom 25% from rest of dataUpper Quartile: separates top 25% from rest of dataIQR = upper quartile – lower quartileContains the middle half of the data

5 Number Summary

Max

Q3 (upper quartile)

Median

Q1 (lower quartile)

Min

Find 5

Number Summary

99, 53, 93 , 82 , 85 , 64 , 75 , 62 , 74 , 81 , 73 , 70 , 81 , 73 , 94, 67 , 93 , 87 , 85 , 36 , 80 , 78

Summarizing Symmetric Distributions: The Mean

When data is skewed or contains outliers, the

median

is a useful measure of the center.

For symmetric data distributions, the

mean is another useful calculation for the center.The mean is the arithmetic average Balancing point for the histogram

Mean vs. Median

There is a rumor that dean of UNC announced that the average starting salary of graduates majoring in geography in 1984 was $300,000. That seems a bit high, any idea why?

Well, it turns out that Michael Jordan was a geography major and got a $3,000,000 contract in the NBA. While the rest of the geography majors made $25,000 - $45,000, this outlier distorted the mean.

Spread: The Standard Deviation

IQR measures spread, but only uses 2 data values.

The

standard deviation

uses

every data value.Only makes sense with symmetric data.Measures how far each data value is from the mean and averages them together.

Calculating Std. Dev. by hand

data

deviations

Squared

deviations1010 – 6 = 442= 1644 – 6 = -2(-2)2= 422 – 6 = -4

(-4)

2

= 16

8

8 – 6 = 2

2

2

= 4

6

6 –

6 = 0

0

2

= 0

Data: 10, 4, 2, 8, 6 mean = 6

Sum of squared deviations: 40

Divide by n-1: 10

Square root: ≈ 3.16

Standard Deviation = 3.16

Why do we want to use the Standard Deviation?

Look at these 3 data sets:

0, 0, 0, 0, 0, 10, 10, 10, 10, 10

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

0, 0, 0, 5, 5, 5, 10, 10, 10

Find their mean, median, mode and spread. What do you see?

Looking at Histograms again

Source:

Intro Stats,

DeVeaux

For each of the data sets below, create a histogram and use that to decide which set of summary statistics to calculate and then calculate them using Minitab.

Neck SizesStudent EmailGasoline Usage