来源：蜘蛛抓取(WebSpider)
时间：2011-10-10 07:08
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range approx
Start Quilting 2 - Figuring Your Quilt Yardage
Two --Figuring Your Quilt Yardage!
You've got some planning
to do!! A good investment for this stage of quilting will be a pad of graph paper,
& a good set of colored pencils, and a calculator!
out how much fabric to buy, you will need to know these things:&
What size quilt are you making?
Will it drop down the sides of the
bed?& How far?
Will it tuck
under the pillows?&
How large is each block within the
How will I lay out my blocks?
How wide is the fabric you will be
Am I using
sashing between the rows and blocks?
Quilt Approx. Sizes
Children's
Quilt Approx. Sizes
by 20&-24&&
Newborn baby
you have these decisions made... you can proceed with figuring your yardages!
Let's say as an example you have decided to make a
FULL size Quilt. The pattern you are using& makes a 12& block. Using the
APPROX. Quilt Size chart above you see that to make a quilt 81& x 88& ....you
will need to make your quilt& about 7 blocks wide by 8 blocks long.
This will give
you a quilt that is actually 84& x 96&& This assumes that you have planned
Block-to-block construction. If you prefer to have sashing between your blocks...you will
need to decide HOW WIDE you want the sashing, and figure that in. You will need to make
less blocks if you chose to have sashing in-between.
decided on all that...now you are ready to start doing the math!! Each block pattern is
comprised of pieces. You will need to figure out how many of each piece you can get out of
a yard of fabric. To do this, it is handy to know the decimal equivalents of each commonly
used fabric fraction...for instance 1/8th, 1/3rd, 3/8ths ...etc...see the table
<font color="#/8 yd=4.5 inches,
1/4=9, 3/8=13.5, 1/2=18, 5/8=22.5, 3/4=27, 7/8=31.5, 1
<font size="3" color="#/2
<font size="3" color="#/8
<font size="3" color="#/4
<font size="3" color="#/8
<font size="3" color="#
<font size="3" color="# 1/8
<font size="3" color="# 1/4
<font size="3" color="# 3/8
<font size="3" color="# 1/2
<font size="3" color="#.0
<font size="3" color="#.125
<font size="3" color="#.25
<font size="3" color="#.375
<font size="3" color="#.5
&Plan on most fabrics
being 42& wide after you remove the unusable selvedge edges...and a yard is
36&long...
I'll walk you through the process of calculating
one quilt, using a 12& Starflower Block as the
Example.&&&& The Supplies List for that
pattern tells you that you need the following amounts for one
1 needs: two strips 3-7/8& x 7-3/4&
2 needs: two strips 3-7/8& x 7-3/4&
3 needs: one strip 3-1/2& x 14& AND two
strips 3-7/8& x 7-3/4&
If we are making a FULL sized quilt, and
doing Block-to-Block construction (No Sashings) we will plan on
making 56 blocks, so will need 56 of each of the fabric units
needed. Our job is to figure out how many yards that means! There
are just FIVE STEPS:
Step One: First, multiply
the strips/units you know you need( shown above in the Supplies
List) times the number of blocks you plan to make ( 56):
For Color 1,& two
strips 3-7/8& x 7-3/4& times 56 = we
will need 112 strips 3-7/8& x 7-3/4&&
For Color 2,&
two strips 3-7/8& x 7-3/4& times 56 = we
will need 112 strips 3-7/8& x 7-3/4&&
For Color 3, one strip 3-1/2&
x 14& times 56 = we will need 56 strips 3-1/2& x
14&&&&&&&&&&&&&&
AND two strips 3-7/8& x
7-3/4& times 56 = we will need
112 strips 3-7/8& x 7-3/4&&
Step Two: Next, working with
one color at a time, find out how many full length strips/units
you can get from each 42& width of fabric you cut:
For Color 1,& 42&
divided by 7-3/4& = 42divided by 7.75 = 5.42...so call it
FIVE (as .42 of a unit won't be long enough to use) (Always round
DOWN on this part!)
For Color 2,& 42&
divided by 7-3/4& = 42/7.75 = 5.42...so call it FIVE (as .42
of a unit won't be long enough to use) (Always round DOWN on this
For Color 3,& 42&
divided by 14& = 3-THREE ( this came out exactly-we love
that!) for the first unit& AND&& for the second
Color 3 strip, 42& divided by 7-3/4& = 42/7.75 =
5.42...so call it FIVE (as .42 of a unit won't be long enough to
use) (Always round DOWN on this part!)
Step Three: Next, still
working with one color at a time, divide the number of units you
NEED, by the number of units you can get from each strip you cut:
For Color 1, You NEED 112&
(3-7/8& x 7-3/4&) strips . You are going to get FIVE
from each 3-7/8& wide strip you cut... so 112 divided by 5 =
22.4& Call it 23 strips of that width you will need to
cut.( Always round UP on this part!)
For Color 2, You NEED 112&
(3-7/8& x 7-3/4&) strips . You are going to get FIVE
from each 3-7/8& wide strip you cut... so 112 divided by 5 =
22.4& Call it 23 strips of that width you will need to
cut.( Always round UP on this part!)
For Color 3, You NEED 56
(3-1/2& x 14&) strips. You are going to get THREE from
each 3-1/2& wide strip you cut. 56 divided by 3 = 18.6 Call
it 19 strips of that width you will need to cut AND you
NEED& 112& (3-7/8& x 7-3/4&) strips . You are
going to get FIVE from each 3-7/8& wide strip you cut... so
112 divided by 5 = 22.4& Call it 23 strips of that
width you will need to cut.( Always round UP on this part!)
Step Four: Now, still
working with one Color at a& time, MULTIPLY the number of
Strips you are going to cut times the width of each strip to find
out how many inches of fabric that is:
For Color 1, 23 strips x 3.875 =
89.125&, or 89 1/8&. Find the nearest real measurement =
90 inches of Color 1 you need to get your units
For Color 2, 23 strips x 3.875 =
89.125&, or 89 1/8&. Find the nearest real measurement =
90 inches of Color 2 you need to get your units
For Color 3, 19 strips x 3.5 =
66.5&& Find the nearest real measurement = 67 inches of
Color 3 you need to get your first units, AND 23 strips x 3.875 =
89.125&, or 89 1/8&. Find the nearest real measurement =
90 inches of Color 1 you need to get your second units. 67 plus 90
= 157 inches to get all units needed.
Step Five: And now, still
working with one color at a time, DIVIDE the number of inches you
need by 36& ( one Yard) to find out how many YARDS you need:
For Color 1, 90& divided by
36& = 2.5 yards
For Color 2, 90& divided by
36& = 2.5 yards
For Color 3, 157& divided by
36& = 4.361& or 4.5 yards
Then when you go to buy your fabric you
may want to bump each figure UP a little to plan for errors, or
whatever. Add more if you want to make BINDING. Add more if you
want to add BACKING. Add more if you absolutely LOVE IT :o)
***special
always round UP when buying yardage...getting a spare 1/4 yard of each
fabric will hurt you far less than running out, and being unable to find the same fabric
again!!!....believe me...you will find ample ways to use the leftovers!! :o)
IN! Through the quilting forum at
I just heard of a
wonderful web tool for you...it is on
site, and is
a FABRIC CALCULATOR...you pop in how many blocks of& unfinished
size, and the approximate percentage of each color, and it pops out
your needed yardages! I can't swear that this works perfectly, but it
does look like a good ROUGH guide! the URL is:
OR... here's
another calculator! This one does more precise calculations based on
thee real number of pieces you are going to need for your quilt!self study - Propagation of error using 2nd-order Taylor series - Cross Validated
to customize your list.
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I am reading a text, "Mathematical Statistics and Data Analysis" by John Rice. We are concerned with approximating the expected value and variance of the random variable $Y$. We are able to calculate the expected value and variance of the random variable $X$ and we know the relation $Y = g(X)$. So, it's possible to approximate the expected value and variance of $Y$ using the Taylor series expansion of $g$ about $\mu_X$.
On page 162, he lists 3 equations.
The expected value of $Y$ using the 1st-Order Taylor series expansion. It is: $\mu_Y \approx g(\mu_X)$. This is referred to later in my question as $E(Y_1)$.
The variance of $Y$ using the 1st-Order Taylor series expansion. It is: $\sigma_Y^2 \approx \sigma_X^2 (g'(\mu_X))^2$. This is referred to later in my question as $Var(Y_1)$.
The expected value of $Y$ using the 2nd-Order Taylor series expansion. It is $\mu_Y \approx g(\mu_X) + \frac12 \sigma_X^2 g''(\mu_X)$. This is referred to later in my question as $E(Y_2)$.
Note that there are two different expressions for $Y$ because we are using two different orders in the Taylor series expansion. Equations 1 and 2 refer to $Y_1 = g(X) \approx g(\mu_X) + (X-\mu_X)g'(\mu_X)$. Equation 3 refers to $Y_2 = g(X) \approx g(\mu_X) + (X-\mu_X)g'(\mu_X) + \frac12 (X-\mu_X)^2 g''(\mu_X)$.
Note that specifically the equation for $Var(Y_2)$ is not given. Later, the author seems to use the equation for the variance of $Y_1$ (Equation 2), when in fact he is referring to expected value of $Y_2$ (Equation 3). This seems to imply $Var(Y_2) = Var(Y_1)$.
I have tried to calculate by hand $Var(Y_2)$, and I am getting a somewhat complicated expression. Here is my work (I stopped because at the end I am getting $X^3$ terms in the expectation):
\begin{aligned}
Var(Y_2) &= E[( g(\mu_X) + (X-\mu_X)a + \frac12 (X-\mu_X)^2 b - g(\mu_X) - \frac12 \sigma_X^2 b
&= E\left[((X-\mu_X)a + \left(\frac12 (X-\mu_X)^2 - \frac12 \sigma_X^2)b\right)^2\right] \\
&= E\left[(ca + \left(\frac12 c^2 - \frac12 \sigma_X^2)b\right)^2\right] \\
& = E[c^2 a^2 + ca(c^2 - \sigma_X^2)b + \frac14(c^2-\sigma_X^2)^2 b^2] \\
& = E[(X^2 - 2X \mu_X + \mu_X^2)a^2 + (X-\mu_X)a((X^2 - 2X\mu_X + \mu_X^2) - \sigma_X^2)b \\
& + \frac14((X^2 - 2X\mu_X + \mu_X^2)-\sigma_X^2)^2 b^2]
\end{aligned}
Note that in the above equations, $a = g'(\mu_X)$, $b = g''(\mu_X)$, and $c = X-\mu_X$. What is $Var(Y_2)$?
Assuming $Y=g(X)$, we can derive the approximate variance of $Y$ using the second-order Taylor expansion of $g(X)$ about $\mu_X=\mathbf{E}[X]$ as follows:
$$\begin{eqnarray*}
\mathbf{Var}[Y] &=& \mathbf{Var}[g(X)]\\
&\approx& \mathbf{Var}[g(\mu_X)+g'(\mu_X)(X-\mu_X)+\frac{1}{2}g''(\mu_X)(X-\mu_X)^2]\\
&=& (g'(\mu_X))^2\sigma_{X}^{2}+\frac{1}{4}(g''(\mu_X))^2\mathbf{Var}[(X-\mu_X)^2]\\
& & +g'(\mu_X)g''(\mu_X)\mathbf{Cov}[X-\mu_X,(X-\mu_X)^2]\\
&=& (g'(\mu_X))^2\sigma_{X}^{2}+\frac{1}{4}(g''(\mu_X))^2\mathbf{E}[(X-\mu_X)^4-\sigma_{X}^{4}]\\
& & +g'(\mu_X)g''(\mu_X)\left(\mathbf{E}(X^3)-3\mu_X(\sigma_{X}^{2}+\mu_{X}^{2})+2\mu_{X}^{3}\right)\\
&=& (g'(\mu_X))^2\sigma_{X}^{2}\\
& & +\frac{1}{4}(g''(\mu_X))^2\left(\mathbf{E}[X^4]-4\mu_X\mathbf{E}[X^3]+6\mu_{X}^{2}(\sigma_{X}^{2}+\mu_{X}^{2})-3\mu_{X}^{4}-\sigma_{X}^{4}\right)\\
& & +g'(\mu_X)g''(\mu_X)\left(\mathbf{E}(X^3)-3\mu_X(\sigma_{X}^{2}+\mu_{X}^{2})+2\mu_{X}^{3}\right)\\
\end{eqnarray*}$$
As @whuber pointed out in the comments, this can be cleaned up a bit by using the third and fourth central moments of $X$. A central moment is defined as $\mu_k=\mathbf{E}[(X-\mu_X)^k]$. Notice that $\sigma_{X}^{2}=\mu_2$. Using this new notation, we have that
$$\mathbf{Var}[Y]\approx(g'(\mu_X))^2\sigma_{X}^{2}+g'(\mu_X)g''(\mu_X)\mu_3+\frac{1}{4}(g''(\mu_X))^2(\mu_4-\sigma_{X}^{4})$$
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