What is Statistics?

• Statistics is the science of collecting, analyzing, and drawing conclusions from data– Descriptive Statistics

• Organizing and summarizing– Inferential Statistics

• Generalizing from a sample to the population from which it was selected

• What kind of data is there?• How can it be graphed for visual

comparison?• How can it be described verbally?• How can it be analyzed numerically?

Describing Data

Data--Types of Variables

Categorical Group or category names w/no order

Eye Color (brown, blue, green)

Quantitative Numerical values (in order, can be averaged, etc.)

Weight (117 lbs, 170 lbs, 253 lbs)

Types of Quantitative (Numerical) Data

Discrete Takes on only certain values

Example: Number of siblings, number of pockets in a pair of jeans, number of free throws made in a season,…

ContinuousTakes on any of an infinite

number of values

Example: Time, Weight, Height, …because of our limitations of measurement accuracy we often round to the nearest second, ounce, inch,…

Describing Univariate DataThe distribution of a variable tells us what values the variable takes, how often it takes those values, and

shows the pattern of variation

• Categorical– Bar graph– Segmented Bar

Graph– Pie chart

• Quantitative– Dotplot– Stemplot (Stem & leaf)– Histogram (Frequency

distribution)– Ogive: Cumulative relative

frequency plot– Boxplot

Bar, Segmented Bar, & Pie Charts

0.52293580.7363184

0.1813665

0.47706420.2636816

0.8186335

0%

20%

40%

60%

80%

100%

Children Women Men

Lost

Saved

0

500

1000

1500

Men Women Children

What’s misleading about this graph?

Source: Marist Institute for Public Opinion

How is this graph misleading?

Describing Data using Summary Features of Quantitative Variables

Center—Location in middle of all data

Unusual features - Outliers, gaps, clustersSpread—Measure of variability, rangeShape—Distribution pattern: symmetric, skewed, uniform, bimodal, etc.

Always CUSS in context!

Dotplot for Univariate Quantitative Data

1.11 Stemplot Answer0 3991 13456778892 0001234556688883 256994 13455795 03596 17 08 3669 3

0 30 991 1341 56778892 00012342 556688883 23 56994 1344 55795 035 596 167 078 38 669 3

(c) The distribution is skewed to the right. The spread is approximately 90 (3 to 93). The center of the distribution is at approximately $28.There are several moderate outliers visible in the split-stem plot; specifically, the five amounts of $70 or more. While most shoppers spent small to moderateamounts of money around $30, a “cluster” of shoppers spent larger amounts ranging from $70 to $93.

a) 1| 9 represents $19 spent at storeb)

Stemplots: Stems & Leaves in order Leave stem blank if no leafSplit stems if too few stems

Back-to-back Stemplot

Babe Ruth Roger Maris

| 0 | 8

| 1 | 3, 4, 6

5, 2 | 2 | 3, 6, 8

5, 4 | 3 | 3, 9

9, 7, 6, 6, 6, 1, 1 | 4

9, 4, 4 | 5 |

0 | 6 | 1

Number of home runs in a season

When comparing data, use comparative language! (higher, more than, etc.)

Histogram of Discrete Data: Rolling a fair six-sided die 300 times

42

54

46 45

5954

0

10

20

30

40

50

60

70

1 2 3 4 5 6

Face of Fair Six-sided Die

Fre

qu

ency

1.14 AnswerHistogram of Continuous Data

• The center is located at 350 ($350,000).

• There appears to be one outlier of $1,103,000.

• The distribution is skewed to the right with a peak in the $200,000s.

• The spread is approximately $1,082,000 ($21,000 to $1,103,000)

• Which bars did the $200,000 and $300,000 salaries go?• Border values always go in the bar on the right!• (First bar is salaries of at least 0 to less than $100,000.)

Histograms on the calculator• Enter data into List1 by going to Stat, 1:Edit• Turn StatPlot on and choose histogram option. Set Xlist to

the list you used to enter in the data.• Choose 1 for Freq or a 2nd list if data is stored in two lists

(values in one, frequency in another)• Press Zoom 9:Statplot to set window to the data initially• Check the window and set reasonable, pretty values of min

& max for both x (values) and y (frequency count). The Xscl will set the width of the bins – make this is a “pretty” number also!

• Then press graph to see the adjusted graph• Press trace to see details of the graph

Histogram of People’s Weights

Data from Histogram

Weight Class Interval Frequency

Relative Frequency

Cumulative Relative Frequency

100 to <120 3 0.038 0.038

120 to < 140 21

140 to < 160 24

160 to < 180 19

180 to < 200 5

200 to < 220 3

220 to < 240 4

Total 79

0.304 0.608

0.241 0.849

0.063 0.912

0.038 0.95

0.051 1.001

1.001

0.266 0.304

Ogive: Cumulative Relative Frequency Graph

Weight (in pounds)

Cumulative Relative Frequency

5 Number Summary

Minimum

Q1 (lower quartile) is the 25th percentile of ordered data or median of lower half of ordered data

Median (Q2) is 50th percentile, or middle number of ordered data (average the two middle numbers if there is an even number of #s)

Q3 (upper quartile) is the 75th percentile of ordered data or median of upper half of ordered data

Maximum

Range = Maximum – minimum

IQR(Interquartile Range) = Q3 – Q1

Outlier Formula: Any point that falls below Q1- 1.5(IQR) or above Q3 + 1.5(IQR) is considered an outlier.

Boxplot – using the 5 # summarySalaries from 1.14 – Enter in calc and press stat, calc, 1-var stats

Min 21

Q1 250

Median 350

Q3 543

Max 1103Check for outliers:

• IQR = Q3 – Q1 = 543-250 =293

• Low boundary: Q1 - 1.5(IQR) = 250 – 1.5(293) = -389.5

no outliers on low end since no salaries are less than this

• High boundary: Q3 + 1.5(IQR) = 543 + 1.5(293) = 982.5

one outlier on high end (1103) since it is higher than 982.5

Max value that’s not an outlier

Scatterplot—Bivariate quantitative dataL

on

gJu

mp

_m

6.0

6.5

7.0

7.5

8.0

8.5

9.0

year1880 1900 1920 1940 1960 1980 2000

Olympics - Mens Field Trends Scatter Plot