Chapter 6 AP Stats vocab

How to Sketch a normal curve that looks normal

-the normal curve is bell-shaped and symmetric around it's mean; start at the middle, and Sketch to the right and left from there -draw the normal model for only 3 standard deviations -the place where the bell shape changes from curving downward to curving back up (inflection point) is exactly one standard deviation away from the mean

A z-score of ___ (plus or minus) or more is rare, and a z-score of ___ or ___ is very rare.

-3 _6 -7

How would you describe the shape of a normal curve?

-bell-shaped and symmetric around its mean; start at the middle and Sketch to the right and left from there -draw it for three standard deviations on each side -inflection point: place where the bell shape changes from curving downward to curving back up; exactly one standard deviation away from the mean

How does standardizing a variable affect the shape, center, and spread of its distribution?

-does not change the shape of a distribution -changes the center by making the mean zero -changes the spread by making the standard deviation 1

How does standardizing affect the distribution of a variable?

-does not change the shape of the distribution of a variable -changes the center by making the mean 0 -changes the spread by making the standard deviation 1

68-95-99.7 rule

-empirical rule -in a normal model, about 68% of the values fall within one standard deviation of the mean, 95% of the values fall within two standard deviations of the mean, and 99.7% of the values fall within three standard deviations of the mean

Evaluate k=1, k=2, and k=3.

-k=1; 1-(1/1) =0 -the distribution is far from normal and none of the values are within one standard deviation of the mean -k=2; 3/4, no Matter how strange the shape of the distribution, at least 75% of the values must be within 2 standard deviations of the mean; normal models expect95% in the 2-standard deviation interval -k=3; 8/9, in any distribution, at least 89% of the values lie within 3 standard deviations of the mean

How does a normal probability plot work?

-plot is roughly a diagonal straight line -deviations from the straight line indicate that the distribution is not normal

What does it mean to standardize data into z-scores?

-shift data by the mean and rescale them by the standard deviation

Describe why standardized units are used to compare values that are measured using different scales, different units, or different populations.

-to use the standard deviation as a ruler to measure statistical distance from the mean -to compare events (to see who wins)

Standardizing

-we standardize to eliminate units -standardized values can be compared and combined even if the orginal variables had different units and magnitudes

When is it appropriate to use a normal model to model a set of data?

-when the distribution is unimodal and symmetric

Is there a difference between the 80th percentile and the top 80%?

-yes -the 80th percentile is the value or score below which 80% of the observations may be found -80% of the values are below you -top 80% means that you are 80/100 for percentage

Statistic

A value calculated from data to summarize aspects of the data -mean and standard deviation are statistics

Standardized value

A value found by subtracting the mean and dividing by the standard deviation

In what way does a z-score give an indication of how unusual a value is?

A z-score of 3 (plus or minus) is rare, and a z-score of 6 or 7 is very rare

What does adding or subtracting a constant to every data value affect?

Adds or subtracts the same constant to measures of position -center, percentiles, min and max increase or decrease by the same constant -leaves measures of spread unchanged (range, standard deviation, IQR)

Normal percentile

Corresponding to a z-score gives the percentage of values in a standard normal distribution found at that z-score or below

In any distribution, at least _____ of the values must lie within _____ standard deviations of the mean.

1-(1/k squared) -+/-

First three rules for working with normal models

1. Make a picture 2. Make a picture 3. Make a picture

What do z-scores measure?

The distance of each data value from the mean in standard deviations -ex: a z-score of 2 tells us the data value is 2 standard deviations above the mean; data values below the mean have negative z-scores

Percentile

The nth percentile of a set of data is the value at which n% of the data is below it -ex: the 20th percentile is the value or score below which 20% of the observations may be found

The larger the z-score is, the more _____ it is.

Unusual

How to compare very different values

Use standard deviation -tells us how the whole collection of values varies

Normal percentiles

Use when the value doesn't fall exactly 1,3, or 3 standard deviations from the mean

Letter z

Used to denote values that have been standardized with the mean and standard deviation

Z-scores

Values that are standardized and commonly denoted with the letter z -(value-mean) divided by the standard deviation

_____ of the data lie between the quartiles.

50%

The best way to tell whether your data can be modeled well by a Normal model is to ___.

Make a picture

Parameters of the model

Mean and standard deviation of a normal model; part of the model; help specify the model -a numerically valued attribute of a model -the values of mew and sigma (cursive m and o with a line to the right) in a normal model are paremeters

Standard deviation

Average distance from the mean

Normal models

Bell-shaped curves -appropriate for distributions whose shapes are unimodal and roughly symmetric -a useful family of models for unimodal, symmetric distributions

The normal model is usually represented with _____.

Greek letters

Don't use a normal model when the distribution is ______.

Not unimodal and symmetric

Don't use the mean and standard deviation when _________.

Outliers are present

Explain the difference between mean and mew.

Mean-average of all data values (add data values up and divide by the count); numerical summary of the data -mew- part of the normal model; doesn't come from the data; is not numerical summary of the data; number chosen to help specify the model (parameter)

Describe two methods for assessing whether or not a distribution is approximately normal.

1)make a histogram 2)use a normal probability plot

Standardized values have no ____.

Units

_____% of all values must be within 5 standard deviations of the mean

96%

Normal probability plot

A display to help assess whether a distribution of data is approximately normal -if the plot is nearly straight, the data satisfy the Nearly Normal Condition

Normal probability plot

A display to help assess whether a distribution of data is approximately normal; is the plot is nearly straight, the data satisfy the Nearly Normal Condition

How does adding or subtracting a constant amount to each value in a set of data affect the mean? Why does this happen?

Mean increases or decreases -occurs because we are shifting the data by the same constant

Mean and standard deviation of z-scores

Mean=zero Standard deviation=1

Using standard deviation as a ruler allows us to_______

Measure the statistical distance fro, the mean -compare values that are measured on different variables -compare values with different scales, with different units, or for different individuals

Rescaling

Multiplying each data value by a constant multiplies both the measures of position (mean, median, and quartiles) and the measures of spread (standard deviation and IQR) by that constant

Standard normal model or standard normal distribution

Normal model with mean zero and standard deviation 1 N (mew, sigma) with mean mew =0 and standard deviation sigma =1 -also called standard normal distribution

Don't ________ in the middle of a calculation

Round your results

What is the ruler of statistics?

Standard deviation

What unit of measurement is used to describe how far a set of values are from the mean?

Standard deviation

How does adding or subtracting a constant amount to each value in a set of data affect the standard deviation?

Standard deviation does not change because the data is just shifted

How does multiplying or dividing a constant amount by each value in a set of data (also called rescaling) affect the standard deviation?

Standard deviation is multiplied or divided by that same constant because the data is rescaled

How does multiplying or dividing a constant amount by each value in a set of data (also called rescaling) affect the standard deviation? Why does this happen?

Standard deviation is multiplied or divided by that same constant because the data is rescaled and because we are looking at the data in different units

Explain the difference between standard deviation and sigma.

Standard deviation- average distance from the mean; numerical summary of data -sigma-part of the normal model; not a numerical summary of the data and does not come from the data; number chosen to help specify the model (parameter)

Summaries of data are called ___.

Statistics; usually written with Latin letters

Explain how to standardize a value.

Subtract the mean from the value, then divide this difference by the standard deviation

Where on the normal curve are inflection points located?

The place where the bell shape changes from curving downward to curving back up -exactly one standard deviation away from the mean

Nearly normal condition

The shape of the data's distribution is unimodal and symmetric -check this by making a histogram or a normal probability plot

Normality assumption

When using the normal model, we assume the distribution of the data is normal; usually not true

Shifting data

When we shift data by adding (or subtracting) a constant to each value, all measures of position (center, percentiles, min, max) increase (or decrease) by the same constant

Shifting

adding a constant to each data value adds the same constant to the mean, the median, and the quartiles, but does not change the standard deviation or IQR