GRAPHS AND NUMBERS

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As we study graphs and numerical summaries, we keep firmly in mind where the data come from

and what we hope to learn from them.

Graphs and numbers are not ends in themselves, but aids to understanding.

VARIABLES

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CATEGORICAL

QUANTITATIVE

Nominal - A, B, C or Fenerbahçe, Galatasaray

Ordinal - A > B > C or Fenerbahçe > Galatasaray

Interval - (B - A) = (C - B) Zero is any other number. e.g. Celsius scale

Ratio - B = n A B is n times as much as A Zero means nothing, nil. e.g. Kelvin scale

ALWAYS ASK

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• What variables were measured?

• What are the units of measurement?

• Are the data coded?

VARIABILITY of AVERAGES

Averages are always less variable than measurements

because chance errors are balanced out!

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The more your measurements, the better your statistics.

IMPORTANCE OF GRAPHS

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Making a graphical display is the first step toward understanding data;

performing well-chosen calculations is the second.

A picture is worth a thousand numbers!

HISTOGRAMSIn EXCEL : Tools Data Analysis Histogram

If Data Analysis is not on the Tools menu then

Tools Add-ins Analysis ToolPAk

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EXCEL Example Histogram

GRAPHS

EXCEL Example Graphs

In EXCEL : Insert Chart ...

Play around with different types of charts.

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MEAN vs MEDIAN

x shown with bar.

Use with no outliers.

Not a resistant statistic of center.

Median - resistant measure of center.

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STEMPLOTS vs BOXPLOTS•Stemplots/histograms show distribution. Use for a single set of data.

•Boxplots show median and spread. Use for comparing sets of data.

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REGULARITY

Measures based on routine methods of statistics (e.g. mean. standard deviation) are generally meaningful

only for distributions of sufficiently regular shape.

Quickly resorting to fancy calculations is the mark of a statistical amateur.

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WHY (n - 1) AND NOT n

xn

xn

x x

nx x x

x nx x

i

n

i

i

n

i n n

i

n

i n

n

i

n

i

1 1

1 1

11

1

1

1

1

can be computed from the first n - 1 valueswhen the mean is known, i.e. is dependent on them.x

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VARIANCE and DEVIATION1/49

avdevn

x xii

n

1

1

variance

variance

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12

1nx x

s

i

i

n

( )

EXCEL Example Deviation

WHY DENSITY CURVE?1/60

Each bag 10 kg

1 meter25 kg

ASSUMPTION of NORMALITYGood for:•Means of samples.•Data from large populations.•Repeated measures.•Random phenomena.

NOT good for:•Individual variables (income, service life, etc).

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ASSESSING NORMALITYFIRST Look at stemplot or histogram for:

Skew. Gaps. Outliers.

SECOND Test for 68 - 95 - 99.5 rule.

THIRD Plot normal quantile plot.

EXCEL Example NQ Plot

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