Displaying Distributions with Graphs

Chapter 1Section 1.1

the science of collecting, analyzing, and drawing conclusions from data

Statistics

Descriptive

Inferential

Main Types of Statistics

the methods of organizing & summarizing data

Descriptive Statistics

involves making generalizations from a sample to a population

Inferential Statistics

the entire collection of individuals or objects about which information is desired

Population

a subset of the population, selected for study in some prescribed manner

Sample

any characteristic whose value may change from one individual to another

Variable

observations on single variable or simultaneously on two or more variables

Data

Who?

What?

Why?

When you see a set of data, ask:

What individuals do the data describe?

How many individuals appear in the data?

Who?

How many variables?

What are the definitions of these variables?

What units?

What?

What is the reason data were gathered?

Why?

Types of Variables

Also called “qualitative”

Identify basic differentiating characteristics of the population

Categorical Variables

Also called “numerical”Observations or measurements that take on numerical values

Makes sense to average these values

Two types-discrete & continuous

Quantitative Variables

Listable set of values

Usually counts of items

Discrete (numerical)

Data can take on any value in the domain of the variable

Usually measurements of something

Continuous (numerical)

Univariate-data that describes a single characteristic of the population

Bivariate-data that describes two characteristics of the population

Multivariate-data that describes more than two characteristics

Classifications by the number of variables

Which category of variables would the following be?

GenderAgeHair colorSmokerSystolic blood pressureNumber of girls in class

Categorical or Quantitative

Types of Distributions4 Common Types

Tells us what values the variable takes and how often it takes these values

One variable may take values that are very close together while others might be spread out

Distribution

Refers to data in which both sides are (more or less) the same when the graph is folded vertically down the middle

Bell-shaped is a special typeHas a center mound with sloping tails

Symmetrical

Refers to data in which every class has equal or approximately equal frequency

Uniform

Refers to data in which one side (tail) is longer than the other side

The direction of the skewness is on the side of the longer tail

Skewed (left or right)

Skewed right Skewed left

Refers to data in which two or more classes have the largest frequency and are separated by at least one other class

Bimodal (multi-modal)

How to describe a graph

Normal, SymmetricalSkewedUniformBimodal

Shape of Graphs

Where the middle of the data falls

3 types of central tendencyMean, median, mode

Center of Graphs

Shows how spread out the data is

Refers to the variability of the data

Range, standard deviation, IQR

Spread

Outliers-value that lies away from the rest of the data

Clusters

Gaps

Unusual Occurrences

1. Bar2. Pie Chart3. Dotplot4. Stem-and-Leaf5. Histogram6. Relative cumulative frequency graph7. Time Plot8. Box & Whisker9. Scatter

Types of Graphs

Must label the axes and title the graph

Scale your axesDraw vertical bar above each category name to a height that corresponds to the count in that category

1. Bar Graphs

Bar Graph (example)The graph below shows the proportion of the female labor force aged 25 and older in the United States that falls into various educational categories. The coding used in the plot is as follows:

1. none-8th Grade 6. bachelor’s degree2. 9th grade-11th grade 7. master’s degree3. high school graduated 8. professional degree4. some college, no degree 9. doctorate degree5. associate degree

Proportion

Educational Attainment (women)

Must include all categories that make up a whole.

2. Pie Chart

3. Dotplot

0 100 200 300 400

Age (months)

Tally marks are made for each set of data.

The data below is graphed on the dotplot on the right.

195 194204 199204 204192 204192 192193 214209 222209

A stem and leaf plot displays data like a bar graph, but we break the data apart into stem and leaf plots.

The data point 30 is broken up into 3 0

stem leaf

The data point for 304 is broken up into 30 4

stem leaf

4. Stem and Leaf Plot

The data set {30, 27, 34, 28, 45, 31, 34, 40, 29}

is represented by:

2: 7 8 93: 0 1 4 44: 0 5

The 2 and 9 represents the data point 29. The stems are the tens place and the leaves are the ones digits. Notice numbers are listed in order from smallest to largest.

Stem and Leaf Plot

Draw a bar graph that represents the count in each class. Leave no horizontal space (unlike a bar graph).

The data below is graphed on the dotplot on the next slide.

195 194 204 199 204 204 192204 192 192 193 214 209 222 209

5. Histogram

Histogram (example)

192 198 204 210 216 222

0

1

2

3

4

5

15 Students

Freq

uenc

y

Age (Months)

Also called an Ogive.

A relative frequency histogram has the same shape and the same horizontal scale as the corresponding frequency histogram. The difference is that the vertical scale measures the relative frequencies (measured as a percentage), not frequencies.

6. Relative Cumulative Frequency Graph

Relative Cumulative Frequency Graph

192 198 204 210 216 222

0

1

2

3

4

5

15 Students

Freq

uenc

yAge (Months)

The frequency needs to be changed to percentages.

Place the time on the x axis. Time is the explanatory variable.

Place the observations on the y axis. The observations represent the response variable.

7. Time Plot

Time Plot (example)

These will be discussed later.Box and Whisker-Section 1.2Scatter Plot-Section 3.1

8. Box & Whisker/9. Scatter Plot

The distribution of a variable tells us what values it takes and how often it takes these values.

To describe a distribution, begin with a graph. Bar graphs and pie charts display the

distributions of categorical variables. These graphs use the counts or percents of the categories.

Stem & Leaf plots and histograms display the distributions of qualitative variables. Stem & Leaf Plots separate each observation into a stem and a one-digit leaf. Histograms plot the frequencies (counts) or percents of equal-width classes of values.

Summary

When examining a distribution, look for the shape, center and spread, and for clear deviations from the overall shape. Some distributions have simple shapes, such as symmetric or skewed. Others may be bimodal (more than one major peak).

Outliers are observations that lie outside the overall pattern of a distribution. Always look for outliers and try to explain them.

Summary continued…

A relative frequency graph (ogive) is a good way to see the relative standing of an observation.

When observations on a variable are taken over time, make a time plot that graphs horizontally and the values of the variable vertically. A time plot can reveal trends (patterns) or other changes over time.

Summary continued

On page 64 in your textbook, complete exercises 1.13 and 1.16.

Assignment