Unit 5 sec 3.2 simplifying two or more terms

Unit 5 sec 3.2 simplifying two or more terms

20 December 2015

22:41

To simplify an expression with 2 or more terms, you need to simplify each turn individually, in the way that you have seen. Before you can do that you need to identify which bits of the expression belonging to which term.

Each term after the 1^st starts with a plus or - that is not inside brackets.

Example 9 identifying terms

Mark the terms in the following expressions

a)  -2a - (5a²) + (-4a)

b)  2x × 4xy – 2y × (-5x)

Solution

a)  Begin by marking the start of the 1^st term                 -2a - (5a²) + (-4a)

extended the line under the 1^st term until you reach a plus or minus sign that is not inside brackets. That is the start of the next term

-2a - (5a²) + (-4a)

extend the line under the 2^nd term until you reach a plus or - that is not inside brackets. That is the start of the next term

-2a - (5a²) + (-4a)

extend the line under the 3^rd time until you reach a plus or minus sign that is not inside brackets. This time you do not reach 1 you just reach the end of the expression. So this expression has 3 terms

-2a - (5a²) + (-4a).

b)  Mark the start of the 1^st term

2x × 4xy – 2y × (-5x)

when you reach a plus or - that is not inside brackets that the start of the next term

2x × 4xy – 2y × (-5x)

you do not to reach another plus or - that is not inside brackets, so this expression has to terms.

2x × 4xy – 2y × (-5x).

Strategy to simplify an expression with more than one term

1)  identify the term. Each term after the 1^st starts with a plus or - that is not inside brackets.

2)  Simplify each term, using this strategy on page 22. Include this sign (plus or minus) at the start of each term.

3)  Any like terms.

Example 10 simplifying expressions with more than one term.

Simplify the following expressions.

(A)       -2a - (-5a²) + (-4a)

(B)      2x × 4xy – 2y × (-5x)

Solution

(A)       1^st, identify the terms . Then simplify each term individually. Finally, collect like terms

-2a - (-5a²) + (-4a) = -2a + 5a² - 4a = 5a² - 6a

this could also be written as -6a + 5a²

(B)      identifying the terms, then simplify each term individually. Finally, check for like terms  - there are non here

-2x × 4xy – 2y × (-5x) = 8x²y + 10y

Unit 5 sec 3.1 simplifying terms and simplifying single term

Unit 5 sec 3.1 simplifying terms and simplifying single term

17 December 2015

00:01

Sometimes the terms in an expression need to be simplified, to make the expression easier to work with, and to make it easy to recognise any right terms.

  Term consists of numbers and letters are multiplied together, then it should be written with a coefficient 1^st, followed by the letters. It is often useful to write the letters in alphabetical order. For example, 3B²DA as 3AB²D. This can help you to identify right terms in a complicated expression.

Term includes a letter multiplied by itself, then index notation should be used. For example,

P × P should be simplified to P²

And

P × P × P should be simplified to P³

Example 6 simplifying terms

write the following terms in the shortest forms.

(A)       3 × c × g ×4 ×b

(B)      B × a × 5 × b × b

Solution

(A)       3 × c × g ×4 ×b = 12bcg

(B)      B × a × 5 × b × b = 5ab³

When you simplify a term you should normally use index notation only for letters, not for numbers. For example

3 × 3 × a should be simplified to 9a, not 3²a.

Example 7 multiplying powers

trying to the following term in his shortest form:

2AB × 3AB.

Solution

2AB × 3AB = 2 × 3 ×a 5+4b1+7 = 6a9b8

With multiplying or dividing:

2 signs the same give a plus sign

2 different signs and minus sign.

Example 8 simplifying terms involving minus signs

write the following terms in the shortest forms

a)  4q × (-2p)

b)  -B³ × (-5B)

c)   -A × (-B) × (-A)

Solution

a)  a positive times in negative games and negative            4q × (-2p) = 4q × 2p = -8pq

b)  a negative times negative gives a positive                  -B³ × (-5B) = + b³ × 5b = +5b4

c)  the 1^st negative times the 2^nd negative gives a positive, then positive times the 3^rd negative gives a negative

-A × (-B) × (-A) = -a × b × a = -a²b

Strategy to simplify a term

1)  finding the overall sign and write it at front.

2)  Simplify the rest of the coefficient and write it next.

3)  Write the letters in alphabetical order (usually), using index notation as appropriate.

Expressions can contain terms of the form

+ (-something) or – (-something).

These should be simplified by using the following facts.

v Adding the negative of something is the same as subtracting the something.

v Subtracting the negative of something is the same as adding the something.

Unit 5 sec 2.3 collecting like terms

Unit 5 sec 2.3 collecting like terms

16 December 2015

23:27

Let us look worse. That how it works with numbers. If you have 2 batches of 4 dots, then altogether you have

2 + 3 batches of 4 dots, that is, 5 batches of 4 dots.

Of course, this does not work, just with batches of dots. For example

2 × 7 + 3 × 7 = 5 × 7.

In fact, no matter what number a is,

2a +3a =5a.

This gives you a way to simplify expressions that contain a number of batches of something, added to another number of batches of the same thing. For example, consider the expression

5bc + 4bc.

Adding 4 batches of bc to find batches of bc gives 9 batches of bc:

5bc +4bc = (5 + 4)bc = 9bc.

Terms that are “batches of the same thing” I called like terms. 4 turns to be away terms, the letters and the powers of the letters each term must be the same. So, for example

7√A and 3√A R White term because they are both terms in √A; 2x² and -0.5x² I like terms because they are both terms in x².

However,

5c and 4c² are not like terms because 5c is a term in c and 4c² is a term in c².

Like terms can always be collected in a similar way to the examples above: you just add the coefficients (including -ones). You can add any number of white terms.

Example 2 collecting like terms

simplify the following expressions.

(A)       12m +15m -26m

(B)      0.5XY² + 0.1XY²

(C)      5p -p

(D)      1/3 – 2d

Solution

(A)       12m +15m -26m = (12+ 15 – 26)m = 1m =m

(B)      0.5XY² + 0.1XY² = ( 0.5 +0.1)XY² = 0.6XY²

(C)      5p -p = 5p -1p =(5 – 1)p = 4p

(D)      1/3 – 2d = (1/3 – 2)d = (1/3 – 6/3)d = -5/3d

The fractional ceifficients were not converted to approximate decimal values. In algebra, you should work with exact numbers, such as 1/3 and √5, rather than decimal approximations, wherever possible. However, if you are using algebra to solve practical problem, then you may have to use decimal approximations.

It is easier to spot like terms. If you make sure that all the electors in each term are written in alphabetical order.

Example 3. Recognising like terms

simplify the following expressions.

(A)       5st+2st

(B)      -6q²rp + 4prq²

Solution

(A)       5st+2st = 5st +2st = 7st

(B)      -6q²rp + 4prq² = -6q²rp + 4prq² = -2pq²r

The lower and upper case versions of the same letter are different symbols in mathematics. So, for example, 4y and 9Y are not like terms.

Any to constant terms like terms. They can be collected using the normal rules of arithmetic.

Often an expression contains some light terms in some unlike terms. You can simplify the expression by 1^st changing the order of its terms so that the light terms are grouped together, and then collecting the light terms. This leaves an expression in which all the terms are, which cannot be simplified any further. Here is an example

Example 4 collecting more like terms

simplify the following expressions.

(A)       2a +5a – 7a + 3b

(B)      12 – 4pq – 2q +1 – qp -2

Solution

(A)       the light terms, then let them

2a +5b – 7a + 3b =2a -7a +5b +3b = -5a +8b

(B)      write qp as pq, group the like terms, then collect them.

12 – 4pq – 2q +1 – qp -2= 12 – 4pq -2q +1 – pq -2

= 12+ 1 -2 – 4pq -pq – 2q

= 11 – 5pq – 2q

the terms in the final expression can be written in any order. For example, an alternative answer is 11 – 2q – 5pq.

As you become more used to working with expressions, you will probably find that you drink lots of white terms without grouping them together first.

  Sometimes when you collect 2 or more like terms, you find that the result is zero - that is, the terms cancel each other out.

Example 5 terms that Cancel out

simplify the expression

M + 2N +3M – 2N.

Solution

M + 2N +3M – 2N = 4M +0 = 4M

Unit 5 sec 2.2 what is a term

Unit 5 sec 2.2 what is a term

16 December 2015

22:10

Some expressions are lists of things that are all added or subtracted. Here is an example

-2xy +3z -y².

The things that are added or subtracted in an expression of this sort are cold the terms. The terms of the expressions above are

-2xy, +3z and -y².

The plus or minus sign at the start of each term is part of the term. Well, you are getting used to working with terms.

You need it to make sure that the sign at the start of each term is included along with the rest of the term.

If the 1^st term of an expression has no sign, then the term is added to the other terms, so really it has a plus sign - it is just that we normally do not write a plus sign in front of the 1^st term of an expression. For example, if you have the expression

4a +c - 7√b – 5,

then you could write a + at the start and mark the terms. When we discussed the terms of expression, we often omit the plus signs. This is convenient in the same way that it is convenient to write the number +3 as 3. So, for example, we might say that the expression

-2xy +3z - y²

has terms

-2xy, 3z and -y².

We never omit the minus sign! And, of course, we never omit the plus sign of a term when writing the term as part of an expression, unless it is the 1^st term.

There is a useful way to think of the relationship between expressions terms.

An expression is equivalent to the sum of its term

the expression on the run is obtained by adding the term of expression (2) on the left. The 2 expressions are equivalent because subtracting y² is the same as adding naked of y².

 Because the order in which numbers are added as a matter, you can change the order of the terms in an expression however you like, and you will obtain an equivalent expression, as long as you keep each term together with its sign. For example, you can swap the order of the 1^st 2 terms in the expression

-2xy+ 3z - y²

to give

3z – 2xy - y²

are you can reverse the original order of the terms to give

-y² + 3z – 2xy.

All 3 of these expressions are equivalent to each other.

Changing the order of the terms simplify an expression, but some methods for simplifying expressions are easier to apply if you arrange the terms 1^st.

The term in an expression may be just a number, like 4, ½ or -5. If so, we say that it is a constant term, I just a constant for short. For example, the expression

3pq -2 + 5p²

has one constant term, -2.

On the other hand, if a term is of the form

a number × a combination of letters,

then the number is called the coefficient of the term, and we say that the term is a term in whatever the combination of letters is. For example

2xy is coefficient 2 and is a term in xy

-3z has coefficient -3 and is a term in z

2/3c² has coefficient 2/3 and is a term in c².

You may be tempted to think that term is like a and -a do not have coefficient. However, because they are equivalent to 1a and -1a, respectively, they have coefficients 1 and -1, respectively.

Unit 5 sec 2.1 Expression and what is an expression.

Unit 5 sec 2.1 Expression and what is an expression.

16 December 2015

00:22

In this section some terminology used in algebra, and a useful technique - collecting like terms.

  An algebraic expression, or just expression for short, is a collection of letters, numbers and/or mathematical symbols (such as +, -, ×, ÷, brackets, and so on), arranged in such a way that numbers are substituted for the letters, then you can work out the value of the expression.

  To make expressions easier to work with, we concisely in the way you saw earlier in the module. In particular, we usually omit multiplication signs; things that are multiplied. I just rated next to each other instead.

  When you are working with expressions, the following is the key thing to remember.

Letters represent numbers, so the normal rule of arithmetic applied to them in exactly the same way as they apply to numbers.

Example 1 evaluating and expression

evaluate the expression

4x² - 5y

When x = 2 and y = -3

Solution

If x = 2 and y = -3, then

4x² - 5y = 4 × 2² - 5 ×(-3)

= 4 ×4 – (-15)

= 16 +15

= 31

Every expression can be written new ways.

If two expressions are really just the same, written differently, then we say that they different forms of the same expression, or that their equivalent to each other.

  When we write an expression in a different way, we say that where rearranging, manipulating or rewriting the expression. Often, the aim of doing this is to make the expression simpler, as with the formula for the Bakers profit. In this case we say that where simplifying the expression.

 We use equal signs when working with expressions, but expressions do not contain equal signs. For example, the statements

X +x =2x and 1.24n -0.69n +0.55n

Aren’t expressions - their equations. An equation is made up of two expressions, with an equal sign between them.

 The equation is correct for only one value of N (it turned out to be 500). In contrast, the equations

X + x = 2x and 1.24n – 0.69n = 0.55n

are correct for every value of x and n, respectively. Equations like these which are true to all values of the variables, are called identities.

There are similar differences in the use of letters to represent numbers. The letter N represented a particular number -it was just that we did not know what that number was. This type of letter is called an unknown. A letter that represents any number (or any number of a particular type, such as any integer) is called a variable, usually you do not need to think about whether a letter is an unknown or unbearable.

Unit 5 sec 1.4 how to learn algebra

Unit 5 sec 1.4 how to learn algebra

16 December 2015

00:13

Algebraic notation is “the language of mathematics”, and it takes time to earn, like any new language. Allow yourself time to get to grips with, and keep practising the techniques. Remember that any difficulties will often be quickly sorted out if you call your tutor or post a question on the online forum.

They give you the opportunity to practice and become familiar with, each new technique before you meet the next one - this is important, because most of the techniques build on techniques introduced previously.

Do the activities even if they were easy -many students find that there are small gaps of misunderstanding in their skills that they are not aware of until they attend the activities project the answers against the correct answer. You may even find some activities throw new light on techniques with which you are familiar.

Unit 5 sec 1.3 Answering mathematical questions

Unit 5 sec 1.3 Answering mathematical questions

16 December 2015

00:01

  To hand you think clearly about questions like this, it helps to represent the number that you want to find by a letter. Let us use N to represent the number of children who applied.

The next step is to write down what you know about N in mathematical notation. We know that 30% of N is 150. That is

30/100× N = 150

which can be written more concisely as

3/10N = 150.

This is an example of an equation. To answer the question about the school places, you have to find the value of N that makes the waging correct when it’s substituted in. This is called solving the equation.

One way to solve the equation is as follows:

Three – tenths of N is 150,

so one tenth of N is 150 ÷ 3 = 50,

so N is 10 × 50 = 500.

  The advantage of writing the information in a question. As an equation is that it reduces the problem of answering the question to the problem of solving an equation. There are standard algebraic techniques for solving many equations, even complicated ones.

So now you seem just the beginning of what algebra can do. Algebra is used in many different fields, including science, computer programming, medicine and finance. It is used to create formulas so that computer programs can carry out many different tasks, from calculating utility bills to producing images on screens. Models, so that predictions, such as those about the economy and climate change, can be made by solving equations. Algebra allows owners to describe, analyse and understand the world, to solve problems and to certain procedures.

Unit 5 sec 1.2 finding and simplifying formulas

Unit 5 sec 1.2 finding and simplifying formulas

15 December 2015

23:40

Suppose that a baker makes a particular type of loaf. Each loaf cost 69p to make, and is sold for £1.24. The baker sells all the Loaves that he makes.

On a particular day, the baker makes 30 loaves. Let us calculate the profit that he makes from them. The total cost, in £, of making the loaves is

30 × 0.69 =20.70

the total of money, in £, paid for the loaves is

30 × 1.24 =37.20

so the profit in £ is given by

37.20 – 20.70 = 16.50

that is, the profit is £16.50.

  It will not be useful for him to have a formula to help him calculate the profit made from any number of loaves. To obtain the formula, we represent the number of loaves by a letter, say n, and worked through the same calculation as above, but using n in place of 30.

The total cost, in £, of making the loaves is

N ×0.69 =0.69n.

The total number of money, in £, paid for the loaves that customers is

N × 1.24 =1.24n.

So if we represent the profit by £p, then we have the formula

P = 1.24n -0.69n.

 The formula makes it easy to calculate the profit, because you do not need to think through the details of the calculation. You just substitute in the number and do the numerical calculation. This is the advantage of using a formula.

  In fact, the task of calculating the profit can be made even more straightforward. It is possible to find a simpler formula for p, by looking at the situation in a different way. The profit, in £, for each loaf of bread is

1.24 – 0.69 =0.55

so the profit, in £, for n loaves of bread is

n ×0.55 = 0.55n.

So we have the alternative formula

P = 0.55n.

The alternative formula of the profit is better because it is simpler and using it involves calculations.

   In this case, a simpler formula was found by thinking about the situation in a different way. However, it is often easier to find whatever formula you can, and then use algebra to turn it into a simpler form.

  Algebra can also help you to find formulas. The formula that the baker’s profit was obtained directly from the situation that it describes, but it is often easier to obtain formulas by using other formulas that you know already. Algebra is needed for this process and it is also needed to turn the new formula into a simpler form.

Unit 5 sec 1.1 proving mathematical facts

Unit 5 sec 1.1 proving mathematical facts

15 December 2015

23:31

  What is the point of learning algebra? Why is it useful? In this section will see some answers to these questions.

  In activities 1 and 2, you probably found that with both your starting numbers you obtained the number 5 in the third last step (the last step involving mathematical calculations), and so each time you obtain the letter E.

  The idea behind the trade is that the number 5 is obtained in the last mathematical step, no matter what the starting number is. But how can you be sure that the trick works for every possible starting number?

  There’s a way to check this - using algebra. In this unit you will learn that algebraic techniques that are needed and you will see how to use them to check that the trek always works.

   As you saw earlier in the module, a demonstration that a piece of mathematics always works is called a proof. Proofs of mathematical and needing in all sorts of contexts, and algebra is usually the way to construct them.

Unit 4 sec 4.2 Quartiles and interquartile range

Unit 4 sec 4.2 Quartiles and interquartile range

12 December 2015

22:47

15 dogs have been placed above the number line, this session correspond to the weekly earnings of the 15 members of staff. Where values coincide (for example, there are three values of £280), the dots are placed vertically, one above another.

   When you have a small number of values in the datasets, as is the case here, it is quick and easy to create a simple dotplot of the data wipe excess. Usually it provides a useful, intuitive picture of where the values lie, whether there is some bunching of the data to one side of there are symmetrical, and whether they are outliers.

  How then can the problem of the range being unduly affected by this outlier be sold? You might simply decide to ignore this particular untypical value, but that is a somewhat arbitrary decision and not one that can be called a general method, although it is sometimes done.

Alternatively, you might choose to omit, say, the largest and smallest values into the range of the remaining 13 values. This is a better solution, and one that works well in this particular instance, but there were several outliers at either end, the problem will not be solved. In order to be confident that you have dealt with the outlying problem, you really need to exclude a greater number of values at either end.

Introducing the quartiles

The conventional solution, and the one described now, is to exclude the top quarter and bottom quarter of the values and create a new measure of spread that measures the “range” of the middle 50% of the values. They are known as the quartiles - in particular the lower quartile (Q1) and upper quartile (Q3).

  You will probably find out the description quartile is not totally convincing as it rather depends on how we choose the interpret “a quarter of the way through the dataset”. Incidentally, the median, the value that lies halfway through the data, sometimes referred to as Q2, as it is the second quartile.

  The convention when defining what quartile is Q1 and which is Q3 is that the data can be presented in increasing order of size. Then, even and odd sample sizes need slightly different approaches, and there are various ways of coping with this. The method for finding the quartiles described in the following two examples is used on some graphical calculators - it is straightforward and quite easy to perform. These example also show you how to find the measure of spread known as the interquartile range, or IQR. The interquartile range is the difference between the upper and lower quartiles, that is, it is the value Q3 – Q1.

Example 3 finding the lower and upper quartiles: even sample size

find the lower quartile (Q1), the median and the upper quartile (Q3) of the following dataset.

8 3 2 6 4 1 5 7

then find the interquartile range.

Solution

sort data into increasing order. Find the median

in increasing order, the dataset is

1 2 3 4 5 6 7 8.

The median is the mean of the team middle data values, 4 and 5.

Median = 4.5

to find the lower quartile, focus on the lower half of the dataset find the median of the smallest dataset.

The last half of the dataset is 1 2 3 4.

It is median is 2.5.

So Q1 = 2.5.

To find the upper quartile, focus on the upper half of the dataset and find the median of the smallest dataset.

There are half of the dataset is 5 6 7 8.

It is median is 6. 5.

So Q3 = 6.5.

The interquartile range is the difference between the upper and lower quartiles.

The interquartile range is thus

6.5 – 2.5 = 4.

Example 4 finding the lower and upper quartiles: odd sample size

find the lower quartile (Q1), the median and the upper quartile (Q3) of the following dataset:

1 2 3 4 5 6 7

to find the interquartile range.

Solution

first, find the median.

The median is the middle value of the ordered datasets.

Median = 4

to find the lower quartile, ignore the middle data value find the median of the lower “half” of the dataset.

The lower half of the dataset is 1 2 3.

The median is 2

So Q1 = 2.

To find the outer quartile, ignore the middle data value and find the median of the upper “half” of the dataset.

The half of the dataset is 5 6 7.

Its median is 6

So Q3 = 6

the interquartile range is the difference between the upper and lower quartiles.

The interquartile range is thus

6 – 2 =4.

The examples lead to the following strategy finding the quartiles and in quartile range.

Strategy to find the quartiles and interquartile range of dataset

1.    arrange the dataset in increasing order.

2.    Next:

(A) If there is an even number of data values, then the lower quartile (Q1) is the median of the lower half of the dataset, and the upper quartile (Q3) is the median of the upper half of the dataset.

(B) If there is an odd number of data values, throw out the middle data point (which of course has the median value of the dataset). Then the lower quartile (Q1) is the median of the lower half of the dataset, and the upper quartile (Q3) is the median of the upper half of the new dataset.

(3) the interquartile range (IQR) is Q3 – Q1.

As you have seen, when there is an even number of data values, the dataset breaks neatly in her and the quartiles are simply median of these two half sets. The procedure is slightly more complicated. The original dataset contains an odd number of values, as decision needs to be made about what constitutes these half sets. However, the choice of whether or not to include the middle data value is quite arbitrary - some authors include date and others, as we have done here exclude it. Indeed, there are yet other methods of calculation that are different again and all of these may give slightly different answers for the values of the quartiles. With very small datasets like the ones you have been using, these differences may be noticeable, but in a real investigation, where the sizes would be larger, these small differences tend to disappear.

Unit 4 sec 4.1 Range

Unit 4 sec 4.1 Range 

12 December 2015

22:36

 As you have just seen, simple way of estimating spread is to scan along the data to find this marvellous (minimum, all min) and largest (maximum of Max) values. The range is the difference between these 2 values. In other words,

Range = Max –min.

Example 2   calculating the range

work at the data below, which against the distances, in kilometres, travelled by allowing students to attend an open University tutorial.

Distances from home

distance (Km): 12 40 26 4 2 18 66 30 45 12 15

calculate the range of this data

solution

by inspecting the data, the maximum value is 66 km and the minimum value is 2 km.

so the range of these data is

Max – min = 66 km – 2 km = 64 km.

the range is sometimes a rather inadequate major of spread, particularly where there are one or two extreme outliers in the dataset. In the jargon of statistician, the range can be referred to as a “quick and dirty” measure of spread - it is quick and easy to calculate, and sometimes do a useful overall impression.

Unit 4 sec 3.3 Mean versus median

Unit 4 sec 3.3 Mean versus median

11 December 2015

23:11

You might be thinking it is very well being told how to calculate two different measures of location, but which should you use and when? Well, it has already been suggested that there is no simple university applicable answer to the question. But there are a few advantages and disadvantages of one measure compared with the others.

   In practice, both the mean and median are widely used. They give similar results, but can sometimes differ considerably. Typically, when the values of the dataset of the two orders of one or other end of the range of values, there are large differences between the mean and the median. For this reason, the median, rather than the me tends to be used for summarising earnings. Where the values are symmetrically spread, there will be little difference between the values of the two summarises, in which case it will not matter much which one is chosen.

   To sum up this section, we have discussed summarising a dataset by measuring its location, is a number that might be thought of as “average” “typical” or “central” value. Two Particular measures of location were looked at in detail: the mean and the median. The meaning of a set of numbers is found by adding all the numbers together and dividing by how many numbers there are. To find the median, first sort the data in order of size stop. If there is an odd number of data values, the median is the middle value. If there is an even number of data values, the media is defined as the mean of the middle two values. The mean and median are two measures of location that were used as part of an initial study of the probability words dataset. By comparing them, you were able to give support to the notion that the word “probably” seems to indicate a higher degree of likelihood than the words possible.

Unit 4 Sec 3.2 Measuring location

Unit 4 Sec 3.2 Measuring location

11 December 2015

22:25

The most important and useful summary of dataset is a measure of its location, based on some sort of average or typical value. There is no signal and universal most appropriate of locations, but there are various useful measures   that can be chosen, depending on the situation and on the nature of the data. Each measure has its own pros and cons. The three most common measures of location in satatisties textbooks are the mean, the mode and the median.

  The mean which is often just called the average, is probably quite a familiar to you.

Strategy to find the mean

To find the mean of a set of numbers, add all the numbers together and divide by however many numbers there are in the set.

Example 1 Calculating the mean.

Find the mean of the texting times of the five teachers.

Solution

Add together the numbers in the “teacher times “dataset and divide by however many numbers are in the dataset

The mean is

(18+27+31+36+47)/5 = 159/5 = 31.8 seconds.

If so, you obtained a statistical average known as the median. Speaking roughly, the median is the data value that is in the middle when data are arranged in order. A more precise definition is

Strategy to find the median of a dataset.

To find the median of a set of numbers:

¨    Sort the data into increasing or decreasing order

¨    If there is an odd number of data values, the median is in the middle value.

¨    If there is an even number of data values, the median is defined as the mean of the middle two values.

Calculating the median is a lot easier if the data are already sorted. However, this too becomes a long calculation if you have to order a large set of values. So, for anything but the smallest set of data, it seems appropriate to turn to a calculator, spreadsheet or other software to compute these summary values. Fortunately, these are similar summary calculations can also be easily carried out using the dataplotter software.

Using Dataplotter to measure location

There are 4 types of plot available: Dotplot, Boxplot, Histogram and Scatterplot.

Dataplotter processes the information in two ways. First, the values are displayed visually as dotplots, a type of statistical plot where each data value is represented by the position of a dot along the number line below it. Dotplots area useful way of seeing patterns in data at a glance.

The second main outcome of selecting these datasets is that a set of ten key summary values has been automatically calculated and displayed for each dataset.

Unit 4 sec 3.1 Summarizing data: Location and scanning data

Unit 4 sec 3.1  Summarizing data: Location and scanning data

22 November 2015

15:11

In particular, in this section you will look at simple summary measures of what statisticians often call a location of dataset, that is a single number that represents an “average”, typical or central value.

A statement that describes whereabouts lies is a way of describing the dataset’s location.

ª    Summarising the set of data values by a single number that might be thought of as an “average” or “typical” or “central” value.

ª    Comparing sets of data values on basis of their locations, to see which set of tends to have bigger values.

Measuring language

What do people understand in words “possible” and “probable”?

Or more specifically

What numerical values to people attribute to words “possible” and “probable”?

Notice that these are paired data in the sense that each “pair” corresponds to the response of particular students.

Before performing a detailed analysis of any data set, it is always a good idea to scan the data, looking closely at the numbers to see if any patterns or anomalies stand out. 

Unit 4 sec 2.5 Single and paired data

Unit 4 sec 2.5 Single and paired data

22 November 2015

14:56

Looking just at column H, the values are all based on a single measure (weight) and can be described as single data, you could calculate an average: you will more about averages in section 3, were 2 different types of average RX warned alternatively you could measure how widely dispersed. The values are - in other words, whether the values are tightly clustered together. Why waste bright. Finally, you could plot the values to discern the overall pattern visually and you will be shown a number of useful statistical plots in unit 11. The purpose of doing these things would be to try and gain an insight into baby weights in general.

Suppose now that there was a 2^nd sample, birth weight grumpy different set of mothers, the babies in the 2^nd sample of being classed as premature. This is now a two sample, as opposed to one sample dataset. Other examples of making statistical comparisons might include making a comparison between 2 medical treatments or to commercial products. Again, a sample of measures would be taken from each of the results compared. (When making such a statistical comparison, there is no requirement that the 2 samples contain the same number of values, although they could do.)

A statistical question of interest might be held the baby’s weight (in column H) relate to the way of their mothers at the start of their pregnancy (column F). This question links to pieces of information for each of the people in this study - a case of paired data. In statistical investigation terms, this falls under the general heading of seeking a relationship. A number of important statistical ideas are linked to exploring relationships today with paired data.

In particular, how to classify and distinguish different types of data. Primary and Secondary data are terms that identify the data source -  primary data you collect yourself, whereas secondary data are taken from somewhere else.

Unit 4 sec 2.4 Spurious precision

Unit 4 sec 2.4 Spurious precision

21 November 2015

15:25

The values, examples of spurious precision. In the 1^st case, the data value seems to be the result of the person’s age being given as 29 years and 11 months, and 11 months being 11/12= 0.916666 years. It is likely that this value is entered as 29+11/12, which would automatically be displayed in its decimal form. In the 2^nd case, the spurious precision has arisen by conversion, 2 kg, of data measured in different units (pounds).

 Another way that spurious precision can arise is when figures are quoted to a greater number of significant figures than is warranted in the context.

The accused was a council employee who regularly stole a portion of the money she was coming from the fees paid into machines by motorists in car parks. Did you merely keep a careful record of all the money taken and add it all up accurately.

  Generally speaking, it is customary, when analysing data gathered by others, to assume that the claimed precision is justified unless there is definite evidence to the contrary. Displaying data it is frequently unnecessary to retain their full procession. A key principle here is that displayed data should be just precise enough to reveal the key features - offering the reader an answer containing too many significant figures can easily obscure these patterns in mass of numbers. If you are collecting primary data, you should bear in mind the kinds of difficulty with data discussed above in the case of secondary data, ideally you should be able to go back to the original data collector in check with them. Any suspicions you have about the data.