Statistics: standard deviation with the Casio fx-260Solar

Tutoring Math Foundations 11, standard deviation is front and centre right now.  The tutor continues explaining the standard deviation function on various scientific calculators.

Like the Sharp EL-520W, the Casio fx-260Solar requires you to set the mode to use its standard deviation function. On the Casio, it’s MODE . (mode, then decimal point). What I like is that the Casio has the modes printed just under the screen. Look and you’ll see, in blue, . SD. Of course, SD refers to Standard Deviation.

Example: Calculate the standard deviation of the number set {-18,16,0,45,100,32,27}.

Solution: As mentioned above, key in MODE . to get in SD mode. Then, you should see SD in the upper right corner.

To enter the numbers, you use the M+ key at the bottom right. In SD mode, it becomes the DATA key. Look underneath it; you’ll see DATA printed in blue.

To input -18, you actually have to enter 18 first, then +/-. Reason: the Casio fx-260Solar is reverse-entry.

Input -18 M+, 16 M+ and so on. Unlike the Sharp EL-520W, the Casio fx-260Solar does not give you feedback as it accepts the data.

Reaching the end of our list, we key in 27 M+. It’s time to find the standard deviation. Key in SHIFT 8 to get σn. You should get 34.9, which is of course what we got with the Sharp EL-520W. After all, the data set is the same. To get σn-1, key in SHIFT 9. I believe that Foundations 11 prefers σn.

To clear the statistical data before entering a new list, change the mode to COMP by pressing MODE 0, then press MODE . to get back into SD mode. Changing out of SD mode, then back into it, seems to clear the previous list.

The mode for ordinary use is COMP. Once again: to return to COMP mode, press MODE 0

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Statistics: Calculating the standard deviation using the Sharp EL-520W

Tutoring math, you get asked about statistics.  The tutor offers an article which might be of interest to stats students.

Calculating standard deviation by hand can be laborious.  Keeping track of what you’ve entered, making sure the brackets have been put in right…most people (including myself) will unlikely enter everything correctly the first time.  In a future post, I’ll show a way to organize the procedure so you do succeed.  Today, though, I have some better news:  maybe you don’t have to.

I’ve never met a scientific calculator without a standard deviation function.  Today, for instance, we have the Sharp EL-520W.  I don’t own any other Sharp calculators, but I imagine this process will be similar for most Sharps worth between $8 and $25.

Example:  Find the standard deviation of the following list:  {-18,16,0,45,100,32,27}

Solution: With the Sharp EL-520W, you need to go into STAT mode to access the statistical functions.

Press MODE. You’ll see a choice of NORMAL or STAT. Press 1 for STAT mode. You arrive at another list of choices: SD, LINE, or QUAD. Press 0 for SD. You should now be back at the calculation screen, with STAT 0 showing at top left.

We are ready to enter our numbers, as follows:

Type in -18, then the DATA key. Note: the key’s main label is M+; however, if you look at the printing below it, you’ll see the DATA label in white.

You’ll know if it’s working: after you press the data key, you should see DATA SET= across the top. Right now, a 1 should appear, since -18 is your first number.

Next, type in 16, then press the DATA key. You should now see DATA SET= with a 2. The 2 means, of course, that 16 is your second number.

Continue: enter each number, then DATA. After keying in 27, then DATA, you should see DATA SET= with 7. Having entered all seven values in the data set, you are ready to find the standard deviation.

Look at the 6 key. The σx label appears in green just above it to the right. Because σx appears in green, we need to press the green ALPHA key to access the σx function.

Press ALPHA 6 =. You should get the result 34.9.

A few points:

1) You will notice that the 5 key has a green sx label above it to the right. sx means σn-1. If your statistics course prefers σn-1, use ALPHA 5instead of ALPHA 6.

2) Before entering a new data set, you should clear the memory, so that no old data affects your new results. Therefore, press 2nd F MODE to “Clear All.”

3) When you’re done with statistical calculations, you might want to return to the mode you were at previously. Most likely, it was normal mode. Press MODE 0 to get back.

In upcoming posts, I’ll outline the standard deviation procedure for other scientific calculators.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: adding mixed numerals, part II

Tutoring math, you meet adults upgrading for new careers.  The tutor continues with this topic, which is common on entrance tests.

 
If you saw my last post, you’ll recall my talk about adding mixed numerals. We covered the more difficult way last time; now we’ll do the easier way.

Example: add 325 + 467

Step 1: Convert the mixed numerals to improper fractions.

Recall that to do so, you multiply the whole number by the denominator, add the numerator, then put the answer over top the same denominator:

(1)   \begin{equation*}3\frac{2}{5}=\frac{(3)(5)+2}{5}=\frac{17}{5}\end{equation*}

Similarly,

(2)   \begin{equation*}4\frac{6}{7}=\frac{34}{7}\end{equation*}

Our problem is now transformed into

(3)   \begin{equation*}\frac{17}{5} + \frac{34}{7}\end{equation*}

Step 2: Add the improper fractions.

As always when adding fractions, we need common denominators.

(4)   \begin{equation*}\frac{17}{5}\frac{(7)}{(7)}+\frac{34}{7}\frac{(5)}{(5)}=\frac{119}{35}+\frac{170}{35}\end{equation*}

Step 3: Add the numerators, while rewriting the common denominator.

(5)   \begin{equation*}\frac{119}{35}+\frac{170}{35}=\frac{289}{35}\end{equation*}

Step 4: We would reduce if 289 and 35 shared a common factor, but they do not. If desired, we can put the answer back into a mixed numeral. We divide 289 by 35 to get 8 remainder 9. 8 becomes the whole number part of the mixed numeral, while 9 goes back over 35:

(6)   \begin{equation*}\frac{289}{35}=8\frac{9}{35}\end{equation*}

A few points:

1) Although this method is more straightforward than the previous one, it does lead to handling bigger numbers. If big numbers bother you, you’ll likely prefer the previous method.

2) From grade 11 on, you rarely see mixed numerals in academic math; improper fractions are preferred. However, trades math continues to prefer mixed numerals.

3) If a question was posed in mixed numeral form, you are likely expected to give your final answer as a mixed numeral if possible.

I’ll be covering other operations with mixed numerals in future posts.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: mixed numerals and improper fractions

Tutoring middle school math, you encounter mixed numerals. Although mixed numerals are easier to understand than improper fractions, some computations cannot be done from mixed numeral form. The tutor opens the discussion.

An example of a mixed numeral is 234. There is a whole number part and a fraction part. Notice that the whole number part (in this case, 2), times the denominator (in this case, 4), plus the numerator (in this case, 3) gives 11. The denominator stays the same: 4. Then you have 114, which is the same value expressed as an improper fraction.

Imagine now, that we want to perform 325+ 467. Of course, we know that to add fractions, we need a common denominator. We have two choices: 1)add the mixed numerals or else 2)convert them to improper fractions and then add.

Choice 1: Add the mixed numerals

First, get common denominators for the fractions:

(1)   \begin{equation*}3\frac{2}{5}\frac{(7)}{(7)}+4\frac{6}{7}\frac{(5)}{(5)}\end{equation*}

Simplify the fractions

(2)   \begin{equation*}3\frac{14}{35}+4\frac{30}{35}\end{equation*}

Add the whole numbers together and add the fractions together.

(3)   \begin{equation*}7\frac{44}{35}\end{equation*}

We are done, except for one hang-up: we now have an improper fraction: 4435. Dividing 44 by 35, we get 1 remainder 9. Therefore,

(4)   \begin{equation*}7\frac{44}{35}=7+1\frac{9}{35}=8\frac{9}{35}\end{equation*}

In my next post, I’ll follow up with the easier method, which is to convert the mixed numerals to improper fractions, then add.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: mixture problems in one variable

Tutoring math, this method comes up.  The two variable method is more popular, but I’ve always preferred this one.  The tutor offers it “for your entertainment.”

Mixture problems are common in high school math.

Example:

A grocer has two nut mixes.  One is 25% cashews; the other, 50% cashews.  He wants to concoct 10 kg of a mixture that is 35% cashews.  How many kg of each mixture should s/he use?

Solution:

Let x = kg of the 25% mixture.

Then, 10 – x = kg of the 50% mixture.

Now we track the cashews in each by percentage, leading to the total:

(1)   \begin{equation*}0.25x + 0.50(10-x) = 0.35(10)\end{equation*}

Notice that in math, we don’t use percents; rather, we use decimals.  To get the decimal, divide the percent by 100.  Therefore, 25%=0.25, and so on.

Next, we simplify the equation:

(2)   \begin{equation*}0.25x + 5 - 0.5x = 3.5\end{equation*}

(3)   \begin{equation*}-0.25x + 5          = 3.5\end{equation*}

Subtract 5 from both sides:

(4)   \begin{equation*}-0.25x                =3.5 - 5\end{equation*}

(5)   \begin{equation*}-0.25x                =-1.5\end{equation*}

Divide both sides by -0.25:

(6)   \begin{equation*}\frac{-0.25x}{-0.25}=\frac{-1.5}{-0.25}\end{equation*}

Finally,

(7)   \begin{equation*}x=6\end{equation*}

Looking back at our “let” statements (which we always need to write), we recall that x stands for the kg of the 25% mixture. Therefore, we need 6kg of the 25% mixture, and 4kg of the 50% one. The yield will be 10kg that is 35% cashews.

The one variable method is elegant. I’ll do the two variable method in a future post.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: factoring decimal trinomials

Tutoring math, I’ve encountered this novelty the last few years.  The tutor suggests one way to see it.

Consider the following:

Factor x2 + 6.7x – 2.1

An interesting problem. As mentioned here, your normal strategy for factoring an easy trinomial would be to find the numbers that multiply to -2.1, but add to 6.7. Behold:

x2 + 6.7x – 2.1 = (x + 7)(x – 0.3)

When you foil it out, you get x2 -0.3x +7x -2.1 = x2 + 6.7x -2.1

However, the times tables don’t cover the numbers that add to 6.7 and mulitply to -2.1; is there a better way?

There’s a trick I use. I’ll show it now, then explain it later.

1) When given x2 + 6.7x – 2.1, imagine x2 + 67x – 210.

2) Use easy trinomial factoring. If you don’t know the numbers that multiply to make -210 but add to 67, make a list starting with 1:

1×210
2×105
3×70

3×70 does it. We know that one number must be negative and one positive, since they multiply to -210.

70-3=67; 70x-3=-210

x2 +67x -210=(x+70)(x-3)

3) Recalling the original question x2 +6.7x – 2.1, we find that (x+7)(x-0.3) is the solution. Notice that we take the numbers from step 2) and divide both by 10: 70÷10 = 7, and -3÷10 = -0.3

This handy trick will get you through most such decimal factoring questions.  Just knowing the pattern, you don’t really need to know why it works.  For those who still want to know why, I’ll explain it in a future post:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Biology: physical vs chemical digestion

Tutoring biology 12, you cover most of the human systems, including the digestive.  The tutor comments on a distinction that doesn’t come up in everyday life, but is relevant to biology 12.

People commonly know that digestion means the breaking down of food.  Depending on their training or interest, they may realize that the food is separated into useful molecules that are absorbed.  From there, the absorbed molecules are either consumed (for instance, in the case of glucose), or else they become part of the body (as with calcium).

Digestion itself is “broken down” into two types:  physical and chemical.  Physical digestion is the process of breaking bites of food into smaller and smaller pieces. It begins with chewing, then continues in the stomach, where the food is mashed into a soup.

Chemical digestion breaks down larger molecules (that the body can’t absorb) into smaller ones (that it can).  Mainly, it is done by enzymes, each of which reacts with a specific type of food molecule.  For instance, lipase breaks down fat molecules. Enzymes often end in -ase.

While physical digestion obviously starts in the mouth, chemical digestion does as well. Saliva contains salivary amylase, which begins decomposing starch into sugars.

Much more will be said about digestion in upcoming posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: perfect cubes and cube roots

Tutoring math, you notice that while square roots are familiar to most people, cube roots are less comfortable for many.  The math tutor offers some clarification….

A perfect square is a number like 25 or 49 that you can arrive at by multiplying a number by itself:

5×5=25

7×7=49

A perfect cube is a number like 8 or 125 that you can arrive at by multiplying a number by itself and then by itself again:

2x2x2=8

5x5x5=125

The cube root of a number is what you times by itself, then by itself again, to get that number:

∛125 = 5      because      5x5x5=125

∛1000=10    because      10x10x10=1000

You can have other roots as well:

∜81=3          because       3x3x3x3=81

There will be more about roots and powers in upcoming posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: the Cartesian Plane

Tutoring math, you know that everyone needs to learn this.  The math tutor provides an introduction.

Most people are familiar with the idea of a two dimensional grid.  You can indicate specific locations (points) on the grid by saying how far across to go, then how far up or down.

In grade school math, the starting place is the centre of the plane, where the two axes (lines) cross.  That central point is called the origin.  The location of every other point is described relative to the origin.  The line going across is the x axis; the one going up and down is the y axis.  If all the grid lines are showing, the x and y axes will be labeled.

The coordinates of a point are the numbers that tell its location.  They are written in brackets, with the x coordinate first.  Since the x axis runs across, the first coordinate tells how far across to go (from the origin), while the second coordinate tells how far up or down to go. Of course, the coordinates of the origin are (0,0).

Therefore, every point can be thought of as

(across, up or down)

and also as

(x,y)

If the across number is negative, you go left. If it’s positive, you go right. If the “up or down” number is positive, you go up. If it’s negative, you go down.

One last hint: the numbers mean “moves”. To go to (5,-3), for instance, you start at the origin, move five squares right, and then three down. Behold:

That’s your introduction to the Cartesian plane.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

English: alliteration and consonance

Tutoring English, you deal with literary terms.  The English tutor comments on the oft-mentioned alliteration as well as the less commonly known consonance.

Most people know that alliteration is the repetition of an initial consonant sound over several words:  hairy, homely hulk, for instance.  Some people suggest that a repeated initial vowel sound can also constitute alliteration, such as utter uncertainty.  I’ve always been given the impression alliteration only involves consonant sounds, but Coles Literary Terms doesn’t specify.

Consonance – which is much less discussed – is the repetition of a consonant sound at the ends of words. Hard polished wood, with its three consecutive terminal d sounds, gives an example of consonance.

I hope this helps:)

Source: Literary Terms. Coles Publishing. 2009: Toronto, Canada.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.