THE STEM AND LEAF PLOT

Steam and leaf plots are a method for showing the frequency with which certain classes of values occur. You could make a frequency distribution table or histogram for the values, or you can use a stem and leaf plot and let the numbers themselves to show pretty much the same information.

Example: 

This stem and leaf plot represents the predicted high temperatures for New York City in the next 10 days

5│7 represents a predicted temperature of 57 degrees Fahrenheit.

Use this plot to find the median of the data.

With this kind of plot, the data values are already in numerical order, which make it’s much easier to find the median. If we cross out each pair of highest and lowest values until the middle becomes visible, we can find the median. 

For example, the first pair of highest and lowest values is 70 and 57, the second pair is 69 and 59, and so on.

The median of this data is between 64 and 63.

The exact median is calculated by taking the mean of 64 and 63, which equals 63.5

Example 2:

This plot shows the top 20 final times for the two-man bobsled teams at the 2011 winter Olympics

3:65 │ 65 represents a time of 3 minutes 26.65 seconds

As you can see the difference between the gold medal winner and the bronze medal winner is less than 1 second (3:27.51 – 3:26.65 = 0.86 sec)!

Use this plot to find the mean time for the top twenty bobsled teams.

To find the mean we must add each of the times together and divide by 20 scores. Since all of the times start with 3 minutes, when adding them together, we can just add up the seconds, and add the minutes back in at the end.

Now, add back in the 3 minutes we left out in that last step:

               3 min 29.308 sec

Questions that you can try:

1.Sam got his friends to do a long jump and got this results:

2.3, 2.5, 2.5, 2.7, 2.8, 3.2, 3.6, 3.6, 4.5, 5.0

Try to find the means and make the steam and leaf plot

GOODLUCK!😊😊😊

THE BOX AND WHISKERS

The reason we call the two lines extending from the edge of the box whiskers is simply because they look like whiskers or mustache, especially mustache of a cat.

The five points or dot that you see represents the followings starting from left to right.

Lower extreme: the lowest or smallest value in a set of data.

Lower quartile or first quartile: the median of all data below the median.

Median or second quartile: the middle value of the set of data. If there are two values in the middle, the median is the average of two values.

Upper extreme: the biggest value in the set.

Example 1:

Construct a box and whiskers plot for the data set; {5, 2, 16, 9, 13, 7, 10}

First, you have to put the data set in order from greatest to least or from least to greatest.

From least to greatest we get: 2  5  7  9  10  13  16

Since the smallest value in the set is 2, the lower extreme is 2

The greatest value in the set is 16, the upper extreme is 16

Now, look carefully at the set:

         2  5  7  9  10  13  16

You can see that 9 is located right in the middle of the set of data. 

Therefore, 9 is the median

Now to get the lower quartile, you need all data before the median or 9

          2  5  7  9  10  13  16

Red color number right above we show all data before 9, so 2  5  7

Since the value in the middle for the set 2  5  7 is  5, the lower quartile is 5

Finally, to get the upper quartile, you need all data after the median or 9

          2  5  7  9  10  13  16

In right above we show all data after 9, so 10  13  16

Since the value in the middle for the set 10  13  16  is 13, the upper quartile is 13

your graph should look like this

Draw a rectangle or box starting from the lower quartile to the upper quartile. Draw a vertical segment too to represent the median.

Finally draw horizontal segments or whiskers that connect all five dots together.

The box and whiskers plot for {5, 2, 16, 9, 13, 7, 10} is :

 Example 2: 

Questions that you can try:

1.

2.

 

3. 

GOODLUCK!😊😊😊

STANDARD DEVIATION

The Standard Deviation is a measure of how spread out numbers are.

The formula is easy : it is the square root of the Variance. So now you ask, “What is the variance?”

The Variance

Variance is defined as : the average of the squared differences from the Mean.

To calculate the variance follow these steps: 

•  Work out the Mean (the simple average of the numbers)
•  Then for each number: subtract the Mean and square the result (the squared difference)
•  Then work out the average of those squared differences.

Example:

You and your friends have just measured the heights of your cats (in millimeters):

The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.

Find out the Mean, the Variance and the Standard Deviation.

Your first step is to find the mean:

So the mean (average) height is 394 mm.

Now we calculate each cat’s difference from the Mean:

To calculate the variance, take each difference, square it, and then average the result:

So the Variance is 21, 704

And the standard deviation is just the square root of Variance, so:

And the good thing about the Standard Deviation is that it is useful. 

Now we can show which heights are within one standard Deviation (147mm) of the mean:

So, using the Standard Deviation we have a “standard” way of knowing what is normal, and what is extra-large or extra small.

Example 2:

In the case where we have a sample size of 5 pirates, therefore we will be using the standard deviation equation for sample of a population.

Here are the amounts of gold coins the 5 pirates have:

4, 2, 5, 8, 6,

Now, let’s calculate the standard deviation.

Questions that you can try:

1. Consider the following three data sets A, B and C. 

A = {9,10,11,7,13} 

B = {10,10,10,10,10} Find 

C = {1,1,10,19,19} 

a) Calculate the mean of each data set. 

b) Calculate the standard deviation of each data set. 

c) Which set has the largest standard deviation? 

d) Is it possible to answer question c) without calculations of the standard deviation?

2.The frequency table of the monthly slaries of 20 people is shown below. 

salary(in $)	frequency
3500	5
4000	8
4200	5
4300	2

a) Calculate the mean of the salaries of the 20 people. 

b) Calculate the standard deviation of the salaries of the 20 people.

GOODLUCK!😊😊😊

INEQUALITIES

This sign < means is less than. While this sign > means is greater than. In each case, the sign opens towards the larger number.

Example:

2 < 5 which mean 2 is less than 5. 

Equivalently, 5 > 2 which also mean 5 is greater than 2.

These are the two senses of an inequality:

< And >

Symbol	words	example
>	greater than	X + 3 > 2
<	less than	7x < 28
≤	greater than or equal to	5 ≤ x - 1
≥	less than or equal to	2y + 1 ≥ 7

Solving

Our aim is to have X (or any variable) on its own on the left of the inequality sign:

          Something like: x < 5

                      Or: y ≥ 11

How to solve?

Solving inequalities is very like solving equations, we do most of the same things BUT we must also pay attention to the direction of the inequality.

These things do not affect the direction of the inequality:

• Add (or subtract) a number from both sides
• Multiply (or divide) both sides by a positive number
• Simplify a side

Example 1:

3x < 7 + 3

We can simplify 7 + 3 without affecting the inequality

= 3x < 10

BUT these things change the direction of the inequality which “< becomes > “

• Multiply (or divide) both sides by a negative number
• Swapping left and right hand sides

Example 2:

2y + 7 < 12

When we swap the left and right hand sides, we must also change the direction of the inequality 

= 12 > 2y + 7

Example 3:

Solve X + 3 < 7

If we subtract 3 from both sides, we get

X + 3 < 7

X < 7 – 3

X < 4

And that is the solution: x < 4

In other words, x can be any values less than 4.

Questions that you can try:

1. 

 2.

 3. 

                GOODLUCK!😊😊😊

ARITHMETIC PROGRESSION

Arithmetic progression (AP) or arithmetic sequence of numbers in which each term after the first is obtained by adding a constant, d to preceding term. The constant d is called common difference.

An Arithmetic progression is given by a, (a + d), (a + d), (a + 2d), (a + 3d), ….. Where a = the first term, d = common difference

Example:

1, 3, 5, 7, …. Is an arithmetic progression (AP) with a 1 and d = 2

7, 13, 19, 25, …. Is an arithmetic progression (AP) with a = 7 and d = 6

nth term of an arithmetic progression tn = a + (n – 1)d where tn = nth term, a = the first term,  d = common difference

Example 1:

Find 10^th term in the series 1, 3, 5, 7, ….

a = 1

d = 3 – 1

   = 2

10^th term:

T₁₀ = a + (n – 1)d

T₁₀ = 1 + (10 – 1)2

T₁₀ = 1 + 18

T₁₀ = 19

Example 2:

Find 16^th term in the series 7, 13, 19, 25, ….

a = 7

d = 13 – 7

   = 6

16^th term:

T₁₆ = a + (n – 1)d

T₁₆ = 7 + (16 – 1)6

T₁₆ = 7 + 90

T₁₆ = 97

Sum of first n^th terms in an arithmetic progression

Example 3:

Find 4 + 7 + 10 + 13 + 16 + … up to 20 terms

a = 4

d = 7 – 4

   = 3

sums of the first 20 term

= n ÷ 2[2a + (n – 1)d]

= 20 ÷ 2 [2(4) + (20 – 1)3] 

= 10[8 + (19)3]

= 10(8 + 57)

= 650

Questions that you can try:

1. The first term of an arithmetic sequence is equal to 6 and the common difference is equal to 3. Find a formula for the nth term and the value of the 50th term

2. The first term of an arithmetic sequence is equal to 200 and the common difference is equal to -10. Find the value of the 20 th term
  

GOODLUCK!😊😊😊