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Statistical Measures

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Statistical Measures

Mrs. Watkins

AP Statistics

Chapters 5,6

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

Midrange: average of the extremes

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 important numbers in data set:

Min:

Q1:

Med:

Q3:

Max:

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

graphical display of data using 5 number summary

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

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

Resistant Measures:

Non-resistant Measures:

Mean, Midrange:

Median, IQR, Trimmed Mean:

Range: the spread between high and low

Resistant?

IQR (Interquartile Range) :

Resistant?

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.

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.

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

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

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 ____________.

Symmetric if mean = median

Skewed left if mean < median

Skewed right if mean > median

LeftRight

Mean is in the tail of the data

Uniform distribution: allvalues relatively

evenly distributed across interval

Bimodal distribution: two peaks

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:

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

Mean:

Standard Deviation:

Median:

IQR:

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:

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

Quartiles:

Percentiles:

Z Scores:

standardized score

how a single value compares to entire data set

in terms of position in distribution

z=

Compute your z score for height?

Compute your z score for Math SAT?

Compute your z score for IQ?

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

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

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 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.

- 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)

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.

- Check mean and median
2. Make a NORMAL PROBABILITY PLOT—

3. Make a BOXPLOT on calculator.

AVOID using histograms on calculator to check.