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[SOLVED] Making a data list from a table with intervals

karush

Well-known member
Jan 31, 2012
2,719
in creating a list of numbers to find mean, median, mode, range and some other questions with this table

height in meters| frequency
\(\displaystyle 8 \le h < 10\ \ \ \ \ \ \ \ \ \ 6\)
\(\displaystyle 10 \le h < 12\ \ \ \ \ \ \ \ 5\)
\(\displaystyle 12 \le h < 14\ \ \ \ \ \ \ \ 7 \)
\(\displaystyle 14 \le h < 16\ \ \ \ \ \ \ \ 4\)

how do we make a list when you have intervals? or do you just use the number in between like \(\displaystyle 8 \le h < 10\) would be \(\displaystyle \{9,9,9,9,9,9\}\)
 

karush

Well-known member
Jan 31, 2012
2,719
let me ask a different question would the mean of this table be
\(\displaystyle \frac{4+5+6+7}{4} = \frac{11}{2} \)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,779
Hi karush! :)

in creating a list of numbers to find mean, median, mode, range and some other questions with this table

height in meters| frequency
\(\displaystyle 8 \le h < 10\ \ \ \ \ \ \ \ \ \ 6\)
\(\displaystyle 10 \le h < 12\ \ \ \ \ \ \ \ 5\)
\(\displaystyle 12 \le h < 14\ \ \ \ \ \ \ \ 7 \)
\(\displaystyle 14 \le h < 16\ \ \ \ \ \ \ \ 4\)

how do we make a list when you have intervals? or do you just use the number in between like \(\displaystyle 8 \le h < 10\) would be \(\displaystyle \{9,9,9,9,9,9\}\)
Yes. You would use the number in the middle as you suggest.


let me ask a different question would the mean of this table be
\(\displaystyle \frac{4+5+6+7}{4} = \frac{11}{2} \)
So no.
The mean would be \(\displaystyle \frac{6\cdot 9 + 5 \cdot 11 + 7 \cdot 13 + 4 \cdot 15}{6+5+7+4}\).
 
Last edited:

karush

Well-known member
Jan 31, 2012
2,719
The mean would be \(\displaystyle \frac{6\cdot 9 + 5 \cdot 11 + 7 \cdot 13 + 4 \cdot 15}{6+5+7+4}\).
so would the median of this be from

\(\displaystyle \{9,11,13,15\} = 12\)

and the mode be \(\displaystyle 7\) since it has the highest frequency

do I have to start a new OP if I continue to ask more Q on this table?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
...
do I have to start a new OP if I continue to ask more Q on this table?
As long as your additional questions pertain to the data already provided, it is best to ask further questions regarding it here in this topic. :D
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,779
so would the median of this be from

\(\displaystyle \{9,11,13,15\} = 12\)
The median is the height where half is smaller and the other half is taller.
At height 12, you have 6+5=11 people smaller, and 7+4=11 people taller.
So indeed the median is 12.

and the mode be \(\displaystyle 7\) since it has the highest frequency
The mode is the height that occurs most... but no one has height 7. ;)
 

karush

Well-known member
Jan 31, 2012
2,719
The mode is the height that occurs most... but no one has height 7.
so the most frequent is \(\displaystyle 12\leq h < 14\) or 13 for mode.

my next question is standard deviation and variance
from Wikipedia
In statistics and probability theory, standard deviation (represented by the symbol sigma, \(\displaystyle \sigma\)) shows how much variation or dispersion exists from the average (mean), or expected value.

So I would presume \(\displaystyle \sigma\) here is 2 since that is the size of the intervals

I read variance in Wikipedia but not sure if it applies to this table.
so how is variance derived?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,779
so the most frequent is \(\displaystyle 12\leq h < 14\) or 13 for mode.
Right. :)


my next question is standard deviation and variance
from Wikipedia
In statistics and probability theory, standard deviation (represented by the symbol sigma, \(\displaystyle \sigma\)) shows how much variation or dispersion exists from the average (mean), or expected value.

So I would presume \(\displaystyle \sigma\) here is 2 since that is the size of the intervals

I read variance in Wikipedia but not sure if it applies to this table.
so how is variance derived?
Not quite.

Let's start with variance.
In your case the formula is:
$$\text{Variance} = \frac{\sum n_i \times (x_i - \text{mean})^2}{\sum n_i}$$
where $n_i$ is the frequency of each category, $x_i$ is the mid value of each height interval, and $\text{mean}$ is the value you already found.

Variance is often denoted as $\sigma^2$.
Standard deviation (denoted as $\sigma$) is the square root of the variance.
 

karush

Well-known member
Jan 31, 2012
2,719
Let's start with variance.
In your case the formula is:
$$\text{Variance} = \frac{\sum n_i \times (x_i - \text{mean})^2}{\sum n_i}$$
where $n_i$ is the frequency of each category, $x_i$ is the mid value of each height interval, and $\text{mean}$ is the value you already found.

Variance is often denoted as $\sigma^2$.
Standard deviation (denoted as $\sigma$) is the square root of the variance.
by \(\displaystyle \sum n_i \) would this mean \(\displaystyle 6+5+7+4=22\)

If so then
\(\displaystyle \frac{6 \times (6-11.82)^2 +5 \times (5-11.82)^2 +7 \times (7-11.82)^2 +4 \times (4-11.82)^2}{22}=variance\)
or is this composed wrong?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,779
by \(\displaystyle \sum n_i \) would this mean \(\displaystyle 6+5+7+4=22\)
Yes!

If so then
\(\displaystyle \frac{6 \times (6-11.82)^2 +5 \times (5-11.82)^2 +7 \times (7-11.82)^2 +4 \times (4-11.82)^2}{22}=variance\)
or is this composed wrong?
Almost.
But you have substituted the frequencies instead of the mid interval values for $x_i$.
 

karush

Well-known member
Jan 31, 2012
2,719
Almost.
But you have substituted the frequencies instead of the mid interval values for $x_i$.
how this?


\(\displaystyle \frac{9 \times (9-11.82)^2 +11 \times (11-11.82)^2 +13 \times (13-11.82)^2 +15 \times (15-11.82)^2}{22}=11.307\) or \(\displaystyle \sigma^2\)

thus standard deviation would be \(\displaystyle \sqrt{11.307}=3.3626\)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,779
how this?


\(\displaystyle \frac{9 \times (9-11.82)^2 +11 \times (11-11.82)^2 +13 \times (13-11.82)^2 +15 \times (15-11.82)^2}{22}=11.307\) or \(\displaystyle \sigma^2\)

thus standard deviation would be \(\displaystyle \sqrt{11.307}=3.3626\)
Hold on.
Now you have substituted the mid interval values for the freqencies $n_i$.
Check where it says $n_i$ and where it says $x_i$.

Btw, the meaning of variance is the average of the squared deviations from the mean.
 

karush

Well-known member
Jan 31, 2012
2,719
Hold on.
Now you have substituted the mid interval values for the freqencies $n_i$.
Check where it says $n_i$ and where it says $x_i$.

Btw, the meaning of variance is the average of the squared deviations from the mean.
\(\displaystyle \frac{6 \times (9-11.82)^2 +5 \times (11-11.82)^2 +7 \times (13-11.82)^2 +4 \times (15-11.82)^2}{22}=4.60331 \) or \(\displaystyle \sigma^2\)

so if correct then \(\displaystyle \sqrt{4.60331} = 2.14553\) or \(\displaystyle \sigma\)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,779
Yep. That looks right.
 

karush

Well-known member
Jan 31, 2012
2,719
there's still more ??

Number of Data
\(\displaystyle = 22\) assume sum of frequencies


Interquartile range
assume we could go off the intervals

so \(\displaystyle Q_1=10 \ \ Q_2=12 \ \ Q_3=14\)

then \(\displaystyle 14-10=4\)

range
\(\displaystyle 16-8=8\)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,779
there's still more ??

Number of Data
\(\displaystyle = 22\) assume sum of frequencies
Correct.


Interquartile range
assume we could go off the intervals

so \(\displaystyle Q_1=10 \ \ Q_2=12 \ \ Q_3=14\)

then \(\displaystyle 14-10=4\)
You're not supposed to work from the intervals.
$Q_1$ is the height such that 25 percent is below.
Since 25% of 22 persons is 5.5, the $Q_1$ height is somewhere in the interval 8-10, which contains 6 persons.
There can be some discussion where that height actually is when talking about intervals, but let's keep it simple and say that $Q_1=9$, which is the middle of the lowest interval.
Similarly $Q_3$ is the height with 75% below.
Keeping it simple that is the middle of the third interval. So $Q_1=13$.

Anyway, your interquartile range comes out the same.


range
\(\displaystyle 16-8=8\)
Right.
 

karush

Well-known member
Jan 31, 2012
2,719
thanks everyone for your help. it was a new topic for me
sure I'll be back with more