**TOPIC 11: STATISTICS ~ MATHEMATICS FORM 2**

Statistics is the study of the collection, analysis, interpretation, presentation and organization of data.

Statistics helps to present information using picture or illustration. Illustration may be in the form of tables,

Illustration may be in the form of tables, diagrams, charts or graphs.

**Pictograms**

Display Information by pictograms

For example here is a pictograph showing how many apples were sold over 4 months at a local shop.

**Note that:**

The method is not very accurate. For example in our example we can’t show just 1 apple or 2 apples.

Pictures should be of the same size and same distance apart. This helps easy comparison.

The scale depends on the amount of data you have. If the data is huge, then

one image can stand for large number like 100, 1000, 10 000 and so on.

**Bar Charts**

They are also called bar graphs. Is a graphical display of information using bars of different heights.

Horizontal and Vertical Bar Charts

Draw horizontal and vertical bar charts

For example; imagine you just did a survey of your friends to find what kind of movie they liked best.

We can show that on a bar graph as here below:

Horizontal scale: 1 cm represents 1 movie they watched.

Interpretation of Bar Chat

Interpret bar chart

in a recent math test students got the following grades:

And this is a bar chart.

Horizontal scale: 1 cm represents 2 students

**Line Graphs**

These are graphs showing information that is connected in some way. For example change over time.

**Example 1**

you are learning facts about mathematics and each day you do test to see how Good you are.

**Solution**

We need to have a scale that helps us to know how many Centimeter will represent how many facts that you were correct.

Vertical scale: 1 cm represents 2 facts that you were right

Horizontal scale: 2 cm represents 1 day.

Interpretation of Line Graphs

Interpret line graphs

Example 2

The graph below shows the temperature over the year:

**From the graph we can get the following data:**

- The month that had the highest temperature was August.
- The month with the lowest temperature was February.
- The difference in temperature between February and may is (32
^{0}-29^{0})=3^{0}C. - The total number of months that had temperature more than 30
^{0}C was 9.

**Pie Chart**

This is a special chart that uses “**pie slices**” to show relative size of data. It is also called Circle graph.

**Example 3**

The

survey about pupils interests in subjects is as follows: 30 pupils

prefer English, 40 pupils refer French and 50 pupils prefer Kiswahili.

Show this information in a pie chart.

**How to make them?**

**Step 1:** put all you are data into a table and then add up to get a total.

**Step 2:** divide each value by the total and then multiply by 360 degrees to figure out how many degrees for each “pie slice” (we call pie slice a sector) We multiply by 360 degrees because a full circle has a total of 360 degrees.

**Step 3:**draw a circle of a size that will be enough to show all information

required. Use a protractor to measure degrees of each sector. It will

look like the one here below:

Interpretation of Pie Charts

Interpret pie charts

Example 4

**Interpreting the pie charts.**

How many pupils are between 121-130cm tall?

angle of this section is 36 degrees. The question says there are 30

pupils in the class. So the number of pupils of height 121 – 130 cm is:

^{36}/_{360} x 30 = 3

**Frequency Distribution Tables**

**Frequency**is how often something occurs. For example; Amina plays netball twice on Monday, once on Tuesday and thrice on Wednesday. Twice, once and thrice are frequencies.

By **counting frequencies** we can make **Frequency Distribution table.**

Frequency Distribution Tables from Raw Data

Make frequency distribution tables from raw data

•Write how often a certain number occurs. This is called tallying

- how often 1 occurs? (2 times)
- how often 2 occurs? (5 times)
- how often 3 occurs? (4 times)
- how often 4 occurs? (2 times)
- how often 5 occurs? (1 times)

From the table we can see how many goals happen often, and how many goals they scored once and so on.

Interpretation of Frequency Distribution Table form Raw Data

Interpret frequency distribution table form raw data

**Grouped Distribution Table**

This

is very useful when the scores have many different values. For example;

Alex measured the lengths of leaves on the Oak tree (to the nearest cm)

16, 13, 7, 8, 4, 18, 10, 17, 18, 9, 12, 5, 9, 9, 16, 1, 8, 17, 1, 10,

5, 9, 11, 15, 6, 14, 9, 1, 12, 5, 16, 4, 16, 8, 15, 14, 17.

**How to make a grouped distribution table?**

**Step 1**: Put the numbers in order. 1, 1, 1, 4, 4, 5, 5, 5, 6, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18,

**Step 2**: Find the **smallest** and the **largest** values in your data and calculate the **range**.

**Step 3**:

Find the size of each group. Calculate an approximate size of the

group by dividing the range by how many groups you would like. then,

round that group size up to some simple value like 4 instead of 4.25

and so on.

Let us say we want 5 groups. Divide the range by 5 i.e. 17/5 = 3.4. then round up to 4

**Step 4**: Pick a

**Starting value**

that is less than or equal to the smallest value. Try to make it a

multiple of a group size if you can. In our case a start value of

**0**make the most sense.

our case, starting at 0 and with a group size of 4 we get 0, 4, 8, 12,

16. Write down the groups. Include the end value of each group. (must

be less than the next group):

The largest group goes up to 19 which is greater than the maximum value. This is good.

Step 6: Tally to find the frequencies in each group and then do a total as well.

**Done!**

**Upper and Lower values**

Referring our example; even though Alex measured in whole numbers, the data is **continuous**.

For instance 3 cm means the actual value could have been any were

between 2.5 cm to 3.5 cm. Alex just rounded numbers to whole numbers.

And 0 means the actual value have been any where between -0.5 cm to 0.5

cm. but we can’t say length is negative. **3.5 cm** is called **upper real limit **or** upper boundary** while **–0.5 cm** is called **lower** **real limit **or** lower boundary**. But since we don’t have negative length we will just use 0. So regarding our example the lower real limit is 0.

**0**is called

**lower limit**and

**3**is called

**upper limit**.

See an illustration below to differentiate between Real limits and limits.

Class size is the difference between the upper real limit and lower real limit i.e**. class size = upper real limit – lower real limit**

**N**(capital N) to represent the total number of frequencies.

**Class Mark of a class Interval**

is a central (middle) value of a class interval. It is a value which is

half way between the class limits. It is sometimes called mid-point of a

class interval. Class mark is obtained by dividing the sum of the upper

and lower class limits by 2. i.e.

Class mark =

Interpretation of Frequency Distribution Tables

Interpret frequency distribution tables

Example 5

interpretation of frequency distribution data:

total number of cars in the survey:

Cars

in the survey are most likely to have 1 person in them as this is the

tallest bar – 6 of the cars in the survey had one occupant.

**Frequency Polygons**

This is a graph made by joining the middle-top points of the columns of a frequency Histogram

For example; use the frequency distribution table below to draw a frequency polygon.

Solution

In

a frequency polygon, one interval is added below the lowest interval

and another interval is added above the highest interval and they are

both assigned zero frequency. The points showing the frequency of each

class mark are placed directly over the class marks of each class

interval. The points are then joined with straight lines.

**Interpretation of Frequency Polygons**

Interpret frequency polygons

The frequency polygon below represents the heights, in inches, of a

group of professional basketball players. Use the frequency polygon to

answer the following questions:

**Histograms**

Is a graphical display of data using bars of different heights. It is similar to **bar charts**, but a Histogram groups numbers into **ranges (intervals)**. And you decide what range to use.

For

example; you measure the height of every tree in the orchard in

Centimeters (cm) and notice that, their height vary from 100 cm to 340

cm. And you decide to put the data into groups of 50 cm. the results

were like here below:

Represent the information above using a histogram.

In order to draw histogram we need to calculate class marks. We will use class marks against frequencies.

**Scale:**vertical scale: 1 cm represents 5 trees

Interpretation of Histograms

Interpret histograms

The histogram below represents scores achieved by 250 job applicants on a personality profile.

- Percentage of the job applicants scored between 30 and 40 is10%
- Percentage of the job applicants scored below 60 is90%
- Job applicants scored between 10 and 30 is100

**Cumulative Frequency Curves**

Cumulative means “**how much so far**”. To get cumulative totals just add up as you go.

For example; Hamis has earned this much in the last 6 months.

**How to get cumulative frequency?**

But, for February, the total earned so far is Tsh 12 000 + Tsh 15 000 = Tsh 27 000.

for

March, we continue to add up. The total earned so far is Tsh 12 000 +

Tsh 15 000 + Tsh 13 000 = 40 000 or simply take the cumulative of

February add that of March i.e. Tsh 27 000 + Tsh 13 000 = Tsh 40 000.

The rest of the months will be:

The results on a** cumulative frequency table **will be as here below:

**The last cumulative total should math the total of all earnings.**

**Note**: To draw an Orgive, plot the points vertically above the upper real limits of each interval and then **join the points by a smooth curve**. Add real limit to the lowest real limit and give it zero frequency.

Interpretation of a Cumulative Frequency Curve

Interpret a cumulative frequency curve

Interpretation:

**Exercise 1**

2. The following table represent the number of pupils with their corresponding height.