You have found the following ages (in years) of all DATA_POINTS plural_form(animal( 1 ), DATA_POINTS) at your local zoo:
\qquadDATA.join( ",\\enspace " )
What is the average age of the plural_form(animal( 1 )) at your zoo? What is the variance? You may round your answers to the nearest tenth.
Average age:\quad
mean( DATA ) years old
Variance:\quad
sum( SQR_DEV ) / DATA_POINTS years^2
7.5\pm 0.15 are accepted to allow for rounding part-way through
Because we have data for all DATA_POINTS plural_form(animal( 1 ), DATA_POINTS) at the zoo, we are able
to calculate the population mean
(\color{BLUE}{\mu}) and
population variance (\color{PINK}{\sigma^2}).
To find the population mean, add up the values of all DATA_POINTS
ages and divide by DATA_POINTS.
\color{BLUE}{\mu} \quad = \quad
\dfrac{\sum\limits_{i=1}^{\color{GREEN}{N}} x_i}{\color{GREEN}{N}} \quad = \quad
\dfrac{\sum\limits_{i=1}^{\color{GREEN}{DATA_POINTS}} x_i}{\color{GREEN}{DATA_POINTS}}
\color{BLUE}{\mu} \quad = \quad
\dfrac{plus.apply( KhanUtil, DATA )}{\color{GREEN}{DATA_POINTS}} \quad = \quad
\color{BLUE}{MEAN\text{ YEARS_OLD}}
Find the squared deviations from the mean for each animal(1).
x_i
Distance from the mean
(x_i - \color{BLUE}{\mu})
(x_i - \color{BLUE}{\mu})^2
POINT plural_form(YEAR, POINT )
roundTo( 2, POINT - MEAN ) plural_form(YEAR, roundTo( 2, POINT - MEAN ) )
SQR_DEV[ i ] plural_form(YEAR, SQR_DEV[ i ] )^2
Because we used the population mean(\color{BLUE}{\mu}) to compute the
squared deviations from the mean, we can find the variance
(\color{red}{\sigma^2}), without introducing any bias, by simply averaging the
squared deviations from the mean:
\color{red}{\sigma^2} \quad = \quad
\dfrac{\sum\limits_{i=1}^{\color{GREEN}{N}} (x_i - \color{BLUE}{\mu})^2}{\color{GREEN}{N}}
\color{red}{\sigma^2} \quad = \quad
\dfrac{plus.apply( KhanUtil, $.map( SQR_DEV, function( x ) { return "\\color{purple}{" + x + "}"; }) )}
{\color{GREEN}{DATA_POINTS}}
\color{red}{\sigma^2} \quad = \quad
\dfrac{\color{purple}{roundTo( 2, sum( SQR_DEV ) )}}{\color{GREEN}{DATA_POINTS}} \quad = \quad
\color{red}{VARIANCE_POP\text{ plural_form(YEAR, VARIANCE_POP )}^2}
The average animal( 1 ) at the zoo is MEAN year old.The average animal( 1 ) at the zoo is MEAN years old.
The population variance
is VARIANCE_POP year^2.The population variance
is VARIANCE_POP years^2.
You have found the following ages (in years) of DATA_POINTS plural_form(animal( 1 ), DATA_POINTS). The plural_form(animal( 1 )) are randomly selected from the POPULATION plural_form(animal( 1 ), POPULATION) at your local zoo:
\qquadDATA.join( ",\\enspace " )
Based on your sample, what is the average age of the plural_form(animal( 1 ))? What is the variance? You may round your answers to the nearest tenth.
Average age:\quad
mean( DATA ) years old
Variance:\quad
sum( SQR_DEV ) / ( DATA_POINTS - 1 ) years^2
0.75\pm 0.15 are accepted to allow for rounding part-way through
Because we only have data for a small sample of the POPULATION plural_form(animal( 1 ), POPULATION), we are only able
to estimate the population mean and variance by finding the sample mean
(\color{BLUE}{\overline{x}}) and
sample variance (\color{PINK}{s^2}).
To find the sample mean, add up the values of all DATA_POINTS
samples and divide by DATA_POINTS.
\color{BLUE}{\overline{x}} \quad = \quad
\dfrac{\sum\limits_{i=1}^{\color{GREEN}{n}} x_i}{\color{GREEN}{n}} \quad = \quad
\dfrac{\sum\limits_{i=1}^{\color{GREEN}{DATA_POINTS}} x_i}{\color{GREEN}{DATA_POINTS}}
\color{BLUE}{\overline{x}} \quad = \quad
\dfrac{plus.apply( KhanUtil, DATA )}{\color{GREEN}{DATA_POINTS}} \quad = \quad
\color{BLUE}{MEAN\text{ YEARS_OLD}}
Find the squared deviations from the mean for each sample. Since we don't know the
population mean, estimate the mean by using the sample mean we just calculated
(\color{BLUE}{\overline{x}} = \color{BLUE}{MEAN\text{ YEAR_TEXT}}).
Find the squared deviations from the mean for each sample. Since we don't know the
population mean, estimate the mean by using the sample mean we just calculated
(\color{BLUE}{\overline{x}} = \color{BLUE}{MEAN\text{ YEARS_TEXT}}).
x_i
Distance from the mean
(x_i - \color{BLUE}{\overline{x}})
(x_i - \color{BLUE}{\overline{x}})^2
POINT plural_form(YEAR, POINT )
roundTo( 2, POINT - MEAN ) plural_form(YEAR, roundTo( 2, POINT - MEAN ) )
SQR_DEV[ i ] plural_form(YEAR, SQR_DEV[ i ] )^2
Normally we can find the variance (\color{red}{s^2}) by averaging the
squared deviations from the mean. But remember we don't know the real
population mean—we had to estimate it by using the sample mean.
The age of any animal( 1 ) in our sample is likely to be closer to the average of the DATA_POINTS plural_form(animal( 1 ), DATA_POINTS) we looked at instead of the average of all the plural_form(animal( 1 ), POPULATION) in the zoo. Because of that, the squared deviations from the mean we calculated will probably underestimate the actual deviations from the population mean.
To compensate for this underestimation, rather than simply averaging the squared deviations from the mean,
we total them and divide by n - 1.
\color{red}{s^2} \quad = \quad
\dfrac{\sum\limits_{i=1}^{\color{GREEN}{n}} (x_i - \color{BLUE}{\overline{x}})^2}{\color{GREEN}{n - 1}}
\color{red}{s^2} \quad = \quad
\dfrac{plus.apply( KhanUtil, $.map( SQR_DEV, function( x ) { return "\\color{purple}{" + x + "}"; }) )}
{\color{GREEN}{DATA_POINTS - 1}}
\color{red}{s^2} \quad = \quad
\dfrac{\color{purple}{roundTo( 2, sum( SQR_DEV ) )}}{\color{GREEN}{DATA_POINTS - 1}} \quad = \quad
\color{red}{VARIANCE\text{ plural_form(YEAR, VARIANCE )}^2}
We can estimate that the average animal( 1 ) at the zoo is MEAN year old.We can estimate that the average animal( 1 ) at the zoo is MEAN years old.
There is a variance
of VARIANCE year^2.There is a variance
of VARIANCE years^2.