Statistics & Data Analysis Unit 3

Displaying Data…

• Organizes data

• Helps show similarities and differences

Categorical Variables

• Bar charts

• Pie charts

• Two way tables

Quantitative Variables

• Dot plots

• Stem plots

• Box plots

• Histogram

Bar Graph

• Vertical or horizontal bars

• Can show frequency (#) or relative frequency (%)

• Must have spaces between bars

What do you need in a bar graph?

• Title

• Y-axis label

• X-axis label

• Spaces in between bars

Pie Charts

• Always in Relative Frequency (%)

• Compares parts of a whole

• Title

• Key

Two Way Tables

Used to compare observations for two different categorical variables

• Column variable (Top to bottom)

• Row variable (Left to Right)

Marginal distribution

The totals converted to %

Stemplots

• Leaf is the final digit of the number

• Stems are always on the left side of the line, leaves are on the right side

• Stems are ordered least (top) to greatest (bottom)

• All stems must be written

• Spacing is important

• No commas between each leaf

• Always have a key

Back to back stemplots

Compares two sets of data

Splitting Stems

• Used when the data is clustered on one or just a few stems

• 2 stems for every 10 digit or 5 stems for every 10 digits

Histograms

• Observations are placed into bins/bars of equal widths

• Count how many observations are in each bin & draw bar to that height

• Similar to bar chart but there are no spaces between bars/charts

• Bin width can be changed but you need a minimum of 5 bars

• Label the x-axis, y-axis, and title

• Can do % or # on y-axis

Describing Distributions

You must include the

• Shape

• Center

• Spread

SHAPE- symmetric or skewed

Symmetric- left and right sides are approximate mirror images

Skewed Left- majority of the data is on the right side, but trails out to the left

Skewed Right- majority of the data is on the left side, but trails out to the right

SHAPE- modes

Unimodal- one major mode where the data is collected around

Bimodal- two major modes

Multimodal- multiple modes

Uniform- data is flat

SHAPE- unusual features

Outliers- observations away from the main distribution

Gaps- spaces between clumps of data

CENTER- mean or median

Mean- used with symmetric graphs (no outliers)

Median- skewed left/right (has outliers)

SPREAD

Standard deviation- goes with mean

IQR- goes with median

• Always list the range of the data (min,max)

Center

Mean: Average

• only used for symmetrical, no outliers

Median: 

• Put #’s in order from least to greatest

• Find middle number

• If there are two middle numbers, then aveage the two #’s

Quartiles

• The medians of the upper and lower half

• First (Q1) and third (Q3) quartile

Spread

Spread: the spread that goes with the median is the IQR & Range

Range: (min,max) or Range= max-min=#

IQR: 

• The difference between Q3 and Q1

• IQR= Q3-Q1

• Represents the middle 50% of the data

5 number summary

• Minimum

• Q1

• Median

• Q3

• Maximum

• Outliers?

Finding outliers: IQR Test

1. Find 1.5(IQR)

2. Use that #: subtract from Q1/A and Q3/B

• A= Q1-(1.5 times IQR)

• B= Q3+(1.5 times IQR)

3. (A,B) is that acceptable range

4. Any # outside (A,B) is an outlier

Spread: Standard Deviation

• SD goes with the mean

• Decribes how spread out the data is (Variability)

• Symbol: s

• The higher the # is, the more spread out the data

Properties of Standard Deviation

• s=0, when all data points are the same number

• s is always positive

• Standard deviation is affected by outliers

Mean vs. Median in distributions

Symmetric

• mean=median

Skewed Left

• mean < median

Skewed Right

• mean > median

Rules:

• The mean is affected by outliers

• The median is resistant to outliers

Comparing Distributions

• When comparing 2 distributions, you must still… state the shape, center, and spread of both distributions.

• You must also compare the 3 things in each distribution

• Use comparison words: higher, lower, similar, same, wider, etc.

• Cannot compare mean to median

GUIDE:

The shape of D1 is ____ which is similar to/different than the shaoe of D2 which is ____. Both are unimodal.

The center of D1 is the mean/median of ____ which is higher/lower than the mean/median of D2 which is _____. 

The IQR/SD of D1 is _____ which is wider/smaller than the IQR/SD of D2 which is _____. The range of D1 is (_,_) which is wider/smaller than the range of D2 which is (_,_). 

Other, outliers, gaps.

If I ADD an upper outlier, how will it affect the summary stats?

Mean-increase

Median-stay the same

Range-increase

IQR- stay the same

SD- increase

If I ADD an lower outlier, how will it affect the summary stats?

Mean-decrease

Median-stay the same

range-increase

IQR-stay the same

SD-increase