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Warm Up

• Find the area of a parallelogram whose base is 35 in. and the corresponding height is 12in.

2. Find the base of the triangle whose area is 48 in2 and height is 12 in.

3. Find the circumference of a circle with a diameter of 14cm. 4. The 3D figure Octagonal Prism a) How many vertices does it have? b) How many edges does it have?

5. Find the volume of the rectangular prism whose length is 4 in, width 7 in and height 4 in.

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1. Parallelogram is a quadrilateral in which pairs of opposite sides are parallel.

2. The distance between the pairs of opposite sides of a parallelogram is called an altitude of the parallelogram.

3. The opposite sides and opposite angles of a parallelogram are equal.

Area of the parallelogram is equal to the product of one of its bases and the corresponding altitude. i.e. A = b * h

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A triangle is a kind of polygon that has three sides.

The area of a triangle is given by "half of base times height“.

Area =

whereb  is the length of the base.h  is the height of the triangle.

½ x b x h

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Trapezoid is a quadrilateral in which one pair of opposite sides are parallel.

The two parallel sides of a trapezoid are its bases, the two non parallel sides are its legs.

Area of trapezoid, A = 1/2 * h (b1 + b2)

BA

CD

E

F

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• A circle is a set of points in a plane at a fixed distance from a fixed point.

• The fixed point is called the center of the circle

• The perimeter of the circle is called the circumference of the circle

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3D figures are figures which have length, width and height.

In the term 3D, 3 refers to the numbers of dimensions and D refers to dimension

Cylinder

Rectangular Prism

CubeCone Sphere

Square Pyramid

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• Surface Area of a PrismThe surface area of a prism is the sum of the areas ofall the sides of the prism. The formula for the surfacearea of a prism therefore depends on the type of prism.

• Surface Area of a CylinderThe surface area of a cylinder is the sum of the areasof the two bases and the lateral face of the cylinder.

surface area of a cylinder = 2*r2 + 2rh

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The amount of space occupied by an object is called its volume.

The volume of a rectangular prism is given by the formula:

Volume = length × width × height

The volume of a cylinder is given by the formula:

Volume = π(radius)2x height

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Frequency - Introduction

Statisticians collect data and process it in order to produce useful information. Each information collected is an observation. Information collected at the beginning stage consists of only raw data. Before any conclusions can be drawn from such data, it must be condensed and arranged in usable form.

In the raw data, there can be repetitions of observations. The number of times a particular observation repeats itself is called its frequency.

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Example: The marks scored by 10 students in a mathematics test is as follows.

47 55 50 65 47 62 51 55 55 64 In the above example, you can see that the score 55 is repeated 3 times. So, the frequency of 55 is 3

Similarly, the frequency of 47 is 2

Marks Obtained

Frequency

47 2

50 1

51 1

55 3

62 1

64 1

Total 10

Let us distribute the data in the example by finding the frequency of all observations and rearranging them, starting from the lowest to the highest observation (or highest to the lowest observation) in a table as given at the left

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Frequency Distribution

Marks Obtained

Frequency

47 2

50 1

51 1

55 3

62 1

64 1

Total 10

This representation is called as Frequency Distribution Table or Frequency Table of marks of 10 students in a class.

In the table, you can see that sum of all frequencies is equal to the number of observations.

The table is useful, as it gives important features of the raw data at a quick glance.

The score 55 has the maximum frequency of 3.

So, we can immediately say that the average score of students in the math’s test is is 55

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Finding frequencies using tally marks

When the number of observation is limited, counting and finding the frequencies is easy. However, when the number of observations is large, it may not be convenient to find the frequencies by simple counting.

In such cases, Tally Marks are used to find the frequencies.Tallies are usually marked in bunches of five for ease in Counting. The fifth tally in a bunch is usually marked diagonally across the earlier four.

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Example: Write the tally marks for the following frequencies.

a) 1 b) 2 c) 5 d) 7

The table below shows the tally marks for the above frequencies

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Frequency distribution using tally marks

Let us consider an example.

Following are the heights (in cm) of of 30 girls in a class.

140, 140, 160,139, 153, 153, 146, 150, 148, 150, 152, 154, 146, 160, 150,

148, 148, 150, 148, 140, 138, 153, 152, 150, 148, 138, 140, 152, 148, 146

Prepare a frequency distribution for the above data and also the mean.

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• First make a table with four rows. • In the first row, write all the observations(x) i.e

the heights of girls in ascending (or descending) order.

• In the second row, keep putting the tally marks against each observation.

• Once all the observations is exhausted, count the tally marks against each height and write the number in the frequency column.

• Sum up all the frequencies (F), to get total number of observations.

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Mean of a Frequency distribution

To find the mean of a frequency distribution, let us

go through the following steps.

1. Multiply each observation (X) by its frequency (F)

2. Find the sum of the products (F *X) obtained in step 1 and write it in

3. Divide the sum obtained (F *X) by the sum of all frequencies (F) to get the required Mean

Thus we can arrive at the following frequency

distribution table.

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Mean of a Frequency distribution

Sum of (F * X)

Sum of (F)Mean

=

4440

30

== 148

Now we can determine the mean of the frequency distribution of heights of 30 girls using the following formula,

Frequency distribution of the heights of 30 girls

So, the average height of 30 girls is 148 cm

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Grouping of data

Organizing data in a table in terms of frequencies

is known as the ungrouped frequency distribution of

raw data. When the number of observations is quite

large, it becomes necessary to condense the given

data into several groups and obtain a frequency

distribution of the number of observations falling in

each group. In such a case, the data is said to be

grouped and the distribution obtained is called a

grouped frequency distribution.

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Grouped Frequency DistributionGrouped Frequency Distribution

Grouping of data is done by going through the following steps.

1. Find out the difference between the highest and lowest values of the observation. The value obtained is called the Range.

2. Depending upon the range, condense the observations into different groups.Each of these groups is called a Class Interval or a Class.

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Usually the number of classes will be kept between 5 & 15.

3.The difference between the upper and the lower limit of a class is called the Size or Width of the Class Interval. It is obtained by dividing the range by the number of classes. The quotient obtained gives approximate size of a class and a convenient number around the quotient can be chosen.

The lower value of a class interval is called its Lower Class Boundary and upper value of a class interval is called its Upper Class Boundary. The sum of the lower and upper limits of the class divided by two is called the Midpoint of the class or the Class Mark

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Grouped Frequency Distribution - Grouped Frequency Distribution - exampleexample

Example: The weight(in kg) of 25 athletes are given below:

67, 83, 74, 81, 67, 84, 61, 54, 59, 60, 36, 45,

55, 58, 75, 57, 39, 49, 73, 77, 58, 63, 44, 88, 52

Construct a group frequency table

1) First, find the range: The maximum and minimum number in the data are 88 and 36 respectively.So, Range = 88 –36 =52

2) Decide the number of classes:As the range is 52, let us decide that the number of classes be 6.

3) Determine the size of Class: Range/number of classes.Size of the class = 52/6 = 8.67. Let the size be

10

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Ensure that the classes include the minimum and maximum number occurring in the data. Minimum number 36 belongs to the class 35 –44 and the maximum number belongs to the class 85-94.Thus, the following group frequency table can be obtained.

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Stem and Leaf PlotStem and Leaf Plot

Data can be shown in a variety of ways including graphs, charts and tables. A Stem and Leaf Plot is a type of graph which summarizes the shape of a set of data (the distribution) and provides extra detail regarding individual values. Here, the data is arranged by place value.

The digits in the largest place is referred to as the stem and the digits in the smallest place are referred to as the leaf (leaves). The leaves are always displayed to the left of the stem. Stem and Leaf Plots are great organizers for large amounts of information.The main advantage of a stem and leaf plot is that the data are grouped and all the original data are shown.

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To make a stem and leaf plot, each observed value must first be separated into its two parts:

•The stem is the first digit or digits;

•The leaf is the final digit of a value;

•Each stem can consist of any number of digits; but

•Each leaf can have only a single digit

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1) On the left hand side of the page, write down the thousands, hundreds or tens (all digits but the last one). These will be the stems.

2) Draw a line to the right of these stems.

3) On the other side of the line, write down the ones (the last digit of a number). These will be the leaves.

4) Finally put a legend or key on the plot to show what you mean by the numbers on this plot.

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How to draw a Stem and Leaf plot ?

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Suppose the observed values are 5, 40, 278 and 45.7.

The stem and leaf values for these are,

5 --- The stem is 0 and the leaf is 5

40 --- The stem is 4 and the leaf is 0

278 --- The stem is 27 and the leaf is 8

45.7--- The stem is 45 and the leaf is 7.

If the range of values is too great, the number 45.7 can be rounded up to 46 to limit the number of stems

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Let us consider an example to draw a stem and leaf plot.

A teacher asked 10 of her students how many books they had read in the last 12 months. Their answers were as follows:

12, 23, 19, 6, 10, 7, 15, 25, 21, 12 Prepare a stem and leaf plot for these data.

Making a Stem and Leaf plotMaking a Stem and Leaf plot

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The stem and leaf plot for the above set of data will be as shown below.

Key: 2|3 = 23

•Stem 0 represents the class interval 0 to 9;

•Stem l represents the class interval 10 to 19;

•Stem 2 represents the class interval 20 to 29.

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An ordered stem and leaf plot is one whose leaves are arranged in ascending order from left to right. Also, in an ordered stem and leaf plot, leaves (digits) are notseparated with punctuation marks (commas or periods) since each leaf is always a single digit.

Consider the following example for making an ordered stem and leaf plot:

Fifteen people were asked how often they drove to work over 10 working days. The number of times each person drove was as follows:

5, 7, 9, 9, 3, 5, 1, 0, 0, 4, 3, 7, 2, 9, 8

Ordered Stem and Leaf plot

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Ordered stem and leaf plot for the above data is as shown below.

Key: 0|3 = 3

The disadvatage here is that this stem and leaf plot does not give much information about the data. With only one stem, the leaves are overcrowded.

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Suppose in an ordered stem and leaf plot, if the leaves become too crowded, then it might be useful to split each stem into two or more components to give much more information about the data.

Let us split the stems in the previous example as given below.1) You can split the stem 0–9 into two intervals of 0–4 and 5–9. or2) You can also split the stem 0–9 into five intervals: 0–1, 2–3, 4–5,

6–7 and 8–9.

Splitting the stems in an ordered Stem and Leaf

plot

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Key: 0|5 = 5

Stem and leaf plot of 2 intervals

• The stem 0(0) means all the data within the interval 0–4.

• The stem 0(5) means all the data within the interval 5–9.

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•The stem 0(0) means all the data within the interval 0–1.

• The stem 0(2) means all the data within the interval 2-3.

•The stem 0(4) means all the data within the interval 4-5.

•The stem 0(6) means all the data within the interval 6-7.

•The stem 0(8) means all the data within the interval 8-9.

Splitting the stems in an ordered stem and leaf

plot

Stem and leaf plot of 5 intervals

Key:0 | 4 = 4

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An outlier is an extreme value of the data. It is an observation value that is significantly different from the rest of the data. There may be more than one outlier in a set of data.

Sometimes, outliers are significant pieces of information and should not be ignored. Other times, they occur because of an error or misinformation and should be ignored.

Outliers

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Back to Back Stem and Leaf plot

Suppose we have two sets of data to compare. Then we can use a 'back to back' Stem and Leaf Plot.

For example, you want to compare the scores of two sports teams,

Team A 30, 57, 37, 33, 59, 42, 39, 48, 53, 51

Team B 32,45, 58, 59, 32, 45, 43, 54, 56, 58

 Write the Stem and Leaf Plot for the above data.

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The tens column is now in the middle and the ones column is to the right and left of the stem column. You can see that the Team B had more games with a higher score than the Team A.

The Team B had only 2 games with a score of 32. The Team A had 4 games with scores 30, 33, 37 and 39. You can also see that the Team B had the highest score of all i.e 59, compared to the Team A with a 57.

Back to Back Stem and Leaf plot

Key: 4|8 = 48

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Your Turn

1. What is the frequency denoted by these tallies? a) b) c)

2. Find the range of the following scores.23, 25, 22, 19, 22, 25, 26, 18, 20, 23, 21, 24, 16

3. Find the lower limit and upper limits of the class intervalsa) 10 – 12 b) 0- 5, 5 -10

4. Find the size of the class intervalsa) 35 – 39 b) 85-94

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Your Turn

5. Find the class mark of the following classes.a) 20 – 24 b) 50 –59

6. Find the approximate average score from the following frequency distribution table.

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7. Following are the set of data which are the high temperatures recorded for 30 consecutive days. Summarize this data by creating a frequency distribution of the temperatures. 50, 45, 49, 50, 43, 49, 50, 49, 45, 49, 47, 47, 44, 51, 51, 44, 47, 46, 50, 44, 51, 49, 43, 43,49, 45, 46, 45, 51, 46

8. Following is the attendance of grade 8th of a school for 28 days out of 80 students in the class .Make a frequency distribution table taking the first group as 31-40.31, 41, 49, 53, 54, 55, 55, 57, 61, 61, 62, 63, 63, 63, 64, 64, 66, 66, 67, 69, 70, 71, 71, 75, 75, 75, 77, 78

Your Turn

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Your Turn

9. Imagine that your school baseball team scores the following number of home runs in 10 games:

4, 5, 8, 5, 7, 8, 9, 8, 8, 7 Make a frequency distribution table for the above data and

find the mean

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Your Turn

10. Britney is a swimmer training for a competition. The number of

50-metre laps she swam each day for 30 days are as follows:

22, 21, 24, 19, 27, 28, 24, 25, 29, 28, 26, 31, 28, 27, 22,39,20,10, 26, 24, 27, 28, 26, 28, 18, 32, 29, 25, 31, 2

a) Prepare an ordered stem and leaf plot. Make a brief comment onwhat it shows.

b) Redraw the stem and leaf plot by splitting the stems into five-unit intervals. Make a brief comment on what the new plot shows.

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Refreshment time

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Q 1.The marks obtained by 20 students in Science during an half yearly examination is as follows.

48, 59, 64, 34, 59, 36, 73, 64, 79, 39, 59, 62, 67, 59, 91 31, 59, 73, 67, 77 Prepare a frequency distribution table for the

above data and also find the Mean.

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Q 2.Study the following table which shows a frequency distribution of marks of 60 students who appeared in an examination.

With reference to the table, determine

a)The lower limit of the sixth class

b)The upper limit of the fourth class

c) The class mark of the third class

d) The size of the fifth class

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Q 3.The weights (to the nearest tenth of a kilogram) of 30 students were measured and recorded as follows:

59.2, 61.5, 62.3, 61.4, 60.9, 59.8, 60.5, 59.0, 61.1, 60.7, 61.6, 56.3, 61.9, 65.7, 60.4, 58.9, 59.0, 61.2, 62.1, 61.4, 58.4, 60.8, 60.2, 62.7, 60.0, 59.3, 61.9, 61.7, 58.4, 62.2

Prepare an ordered stem and leaf plot for the data. Briefly comment on what the analysis shows?

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1. Observations gathered at the beginning stage constitutes the raw data.

2. Frequency denotes the number of times a particular observation occurs in a given data

3. The difference between the highest and lowest values of the observations is known as the range.

4. Frequency distribution is a table showing the frequencies of the various observations of a data.

5. Tally marks are used to find the frequencies when the number of observations is large.

Let’s summarize what we have learnt today

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6. When the number of observations is very large, the data is usually organized into groups called class intervals and the data obtained is called the grouped data.

7. The lower value of a class interval is called its lower limit and upper value of a class interval is called its upper limit.

8. The difference between the upper and lower class limits is called the width or size of the class interval.

9. The mid value of a class interval is called its class mark.

10. A Stem and Leaf Plot is a type of graph that summarizes the shape of a set of data (the distribution) and provides extra detail regarding individual values.

Let’s summarize what we have learnt today

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11. The digits in the largest place is referred to as the stem and the digits in the smallest place are referred to as the leaf (leaves)

12. In an ordered stem and leaf plot, the leaves are arranged in ascending order from left to right.

13. An outlier is an extreme value of the data. It is an observation value that is significantly different from the rest of the data.

14. One can use a back to back stem and leaf plot while comparing two sets of data.

Let’s summarize what we have learnt today

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