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Statistical Measures. Mrs. Watkins AP Statistics Chapters 5,6. MEASURES OF CENTER. Mean : arithmetic average of all data values population mean : sample mean : Formula : Mode : the most common value in a data set. Median : the middle value in a data set

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statistical measures

Statistical Measures

Mrs. Watkins

AP Statistics

Chapters 5,6

measures of center
MEASURES OF CENTER

Mean: arithmetic average of all data values

population mean:

sample mean:

Formula:

Mode: the most common value in a data set

slide3
Median: the middle value in a data set

Midrange: average of the extremes

slide4
Trimmed Mean: when you find the mean

of data set with a certain percentage of

data values trimmed of the ends of the

distribution

Ex:

5 number summary
5 number summary

5 important numbers in data set:

Min:

Q1:

Med:

Q3:

Max:

Q1, Med, Q3, may not be actual data values

boxplot
BOXPLOT

graphical display of data using 5 number summary

(if outliers shown, called “modified box plot”)

outliers
OUTLIERS

Outliers:

IQR Test for Outliers:

(IQR )(1.5) = multiplier M

Q1 - M = outlier lower bound

Q3 + M = outlier upper bound

If values exceed these bounds, they are outliers

resistance
RESISTANCE

Resistant Measures:

Non-resistant Measures:

Mean, Midrange:

Median, IQR, Trimmed Mean:

measures of spread
MEASURES OF SPREAD

Range: the spread between high and low

Resistant?

IQR (Interquartile Range) :

Resistant?

standard deviation
STANDARD DEVIATION

a measure of the average amount of deviation from the mean among the data values

Population St. Deviation:

Sample St. Deviation:

We generally use sx because we usually do not have entire population.

variance
VARIANCE

the square of the standard deviation

what you get before taking square root

Population Variance:

Sample Variance:

This measure not used much in elementary statistics but you need to know what it is.

coefficient of variance
Coefficient of Variance

measure of how relatively large a st. dev. is

Ex: St. deviation of IQ = 15, Mean 100

St. deviation of height = 3 in, Mean 69

comment on the distribution
“Comment on the distribution”

You now have numbers to support your statements, rather than just graphs.

SHAPE:

OUTLIERS:

CENTER:

SPREAD: how widely does the data vary?

Unusual Features: gaps, clusters

shape
SHAPE

If the mean > median, then data distribution

is skewed ________The mean is in the tail.

If the mean < median, then data distribution

is skewed ________The mean is in the tail.

If the mean ≈ median, then data distribution

is approximately ____________.

shape1
SHAPE

Symmetric if mean = median

skewness
SKEWNESS

Skewed left if mean < median

Skewed right if mean > median

Left Right

Mean is in the tail of the data

other shapes
OTHER SHAPES

Uniform distribution: allvalues relatively

evenly distributed across interval

Bimodal distribution: two peaks

transformations to data
TRANSFORMATIONS TO DATA

What would happen to the statistical measures if each data value had a constant added to or subtracted from it?

Mean:

Standard Deviation:

Median:

IQR:

slide19
What would happen to the statistical measures if each data value had a constant multiplied or divided by it?

Mean:

Standard Deviation:

Median:

IQR:

transformations to data set
TRANSFORMATIONS TO DATA SET

What would happen to the statistical measures if one very low or very high data value was added to the set?

Mean:

Standard Deviation:

Median:

IQR:

measures of position
MEASURES OF POSITION

Give a numerical approximation of where a single data value stands compared to the whole distribution

Quartiles:

Percentiles:

Z Scores:

z scores
Z SCORES

standardized score

how a single value compares to entire data set

in terms of position in distribution

z=

how unusual are you
How unusual are you?

Compute your z score for height?

Compute your z score for Math SAT?

Compute your z score for IQ?

normal model
NORMAL MODEL

shows how data is distributed symmetrically along an interval according to empirical rule

Empirical Rule:

of data within 1 st. deviation of μ

of data within 2 st. deviations of μ

of data within 3 st. deviations of μ

another outlier test
ANOTHER OUTLIER TEST

Using Empirical Rule:

Data values of z > +2 st. deviations away

from mean are mild outliers

Data values of z > +3 st. deviations away

from mean are extreme outliers

normal curve
NORMAL CURVE

a theoretical ideal about how traits/characteristics are distributed

Many human traits are approximately normally distributed such as height, body temp, IQ, pulse

Avoid using “normal” when describing data—say

“approximately normal or symmetric” unless clearly mound-shaped, bell-shaped

normal curve1
NORMAL CURVE

Normal curve—symmetric, mound-shaped

Area under curve=

A z score can be used to establish what % of

the curve is less or more than the z score,

and establish probability of a data value being in that position.

finding percentile probability using normal curve
FINDING PERCENTILE/PROBABILITY USING NORMAL CURVE
  • Calculate z score for data value
  • Use calculator: normalcdf under DISTR

key

Looking for area > z score: normalcdf (z, ∞)

Looking for area < z score: normalcdf (∞, z)

Looking for area between z scores:

normalcdf (z1, z2)

finding cut off scores
FINDING CUT OFF SCORES

If you are given a percentile or probability, and

need to determine the “cut off score”

  • Sketch curve to determine where z scoreis located.

2. Determine if you want area above or below this percentile

3. Use INVNORM on calculator

invnorm(percentile)= z score

  • Use z score formula to solve for x.
does the data fit a normal model
Does the data fit a normal model?
  • Check mean and median

2. Make a NORMAL PROBABILITY PLOT—

3. Make a BOXPLOT on calculator.

AVOID using histograms on calculator to check.

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