Mathematics ECLPE

Key Unit competence: Extend understanding, analysis and interpretation of data arising from problems and questions in daily life to the standard deviation

8.0. Introductory Activity 8

1. During 6 consecutive days, a fruit-seller has recorded the number of fruits sold per type

Which type of fruits had the highest number of fruits sold?

b) Which type of fruits had the least number of fruits sold?

c) What was the total number of fruits sold that week?

d) Find out the average number of fruits sold per day.

2. During the welcome test of Mathematics out of 10 , 10 student-teachers of year one ECLPE scored the following marks: 3, 5,6,3,8,7,8,4,8 and 6. 

a) Determine the mean mark of the class. 

b) What is the mark that was obtained by many students?

c) Compare and discuss about the mean mark of the class and the mark for every student-teacher. What advice could you give to the Mathematics tutor?

d) organize all marks in a table and try to present them in an X-Y Cartesian plane by making x-axis the number of marks and y-axis the number of student-teachers.

8.1 Collection and presentation of grouped and ungrouped data

8.1.1. Collection and presentation of ungrouped data

Activity 8.1.1

1. Observe the information provided by the graph and complete the table by corresponding the number of learners and their age 

CONTENT SUMMARY

Every day, people come across a wide variety of information in form of facts or categorical data, numerical data in form of tables. 

For example, information related to profit/ loss of the school, attendance of students and tutors, used materials, school expenditure in term or year, student-teachers’ results. These categorical or numerical data which is numerical or otherwise, collected with a definite purpose is called data.

Statistics is the branch of mathematics that deals with the collection, presentation, interpretation and analysis of data.

 A sequence of observations, made on a set of objects included in the sample drawn from population, is known as statistical data.

Statistical data can be organized and presented in different forms such as raw tables, frequency distribution tables, graphs, etc. 

Qualitative data

Qualitative data is a categorical measurement expressed not in terms of numbers, but rather by means of a natural language description. In statistics, it is often used interchangeably with “categorical” data.  Categorical data represent characteristics such as a person’s gender, marital status, hometown, or the types of movies they like. Categorical data can take on numerical values (such as “1” indicating male and “2” indicating female), but those numbers don’t have mathematical meaning. It couldn’t add them together, for example.

Quantitative data

Quantitative data is a numerical measurement expressed not by means of a natural language description, but rather in terms of numbers. These data have meaning as a measurement, such as a person’s height, weight, IQ, or blood pressure; or they’re a count, such as the number of stock shares a person owns, or how many pages you can read of your favorite book before you fall asleep.

Numerical data can be further broken into two types: discrete and continuous.

• Discrete data represent items that can be counted; they take on possible values that can be listed out. The list of possible values may be fixed (also called finite); or it may go from 0, 1, 2, on to infinity (making it countably infinite). For example, the number of heads in 100 coin flips takes on values from 0 through 100 (finite case), but the number of flips needed to get 100 heads takes on values from 100 (the fastest scenario) up to infinity (if you never get to that 100th heads). Its possible values are listed as 100, 101, 102, 103, . . . (representing the countably infinite case).

• Continuous data represent measurements; their possible values cannot be counted and can only be described using intervals on the real number line. For example, the exact amount of gas purchased at the pump for cars with 20-gallon  tanks would be continuous data from 0 gallons to 20 gallons, represented by the interval [0, 20], inclusive. You might pump 8.40 gallons, or 8.41, or 8.414863 gallons, or any possible number from 0 to 20. In this way, continuous data can be thought of as being uncountably infinite. 

After the collection of data, tally is used to organize and present in a frequency distribution table. Tally means to count by grouping the number of times an item has occurred. When the data are arranged in this way we say that we have obtained 

the frequency distribution.

Raw data

Data which have been arranged in a systematic order are called raw data or ungrouped data.

For Example, the following are marks out of 20 for 12 student-teachers.

13    10    15    17    17    18

17    17    11    10    17    10

Frequency distribution 

A frequency distribution is a table showing how often each value (or set of values) of the collected data occurs in a data set. A frequency table is used to summarize categorical or numerical data. Data presented in the form of a frequency distribution are called grouped data.

Example 

The following data of marks, out of 20, obtained by 12 student-teachers can be presented in a frequency distribution table. 

13     10     15     17     17     18

17     17     11     10     17     10

The set of outcomes is displayed in a frequency table, as illustrated below:

Cumulative frequency

The cumulative frequency corresponding to a particular value is the sum of all frequencies up to the last value including the first value. Cumulative frequency can also defined as the sum of all previous frequencies up to the current point.

Example:

The set of data below shows marks obtained by student-teachers in Mathematics. Draw a cumulative table for the data.

Solution:

The cumulative frequency at a certain point is found by adding the frequency at the present point to the cumulative frequency of the previous point.

STEM AND LEAF DISPLAYS

Is a plot where each data value is split into a leaf usually the last digit and a stem the other digit. The stem values are listed down, and the leaf values are listed next to them. This way the stem groups the scores and each leaf indicates a score within that group

Example: The mathematical competence scores of 10 student-teachers participating in mathematics competition are as follows:15,16,21,23,23,26,26,30,32,41. Construct a stem and leaf display for these data by using1, 2, 3, and 4 as your stems.

This means that data are concentrated in twenties.

Example: The following are results obtained by student-teachers in French out of 50. 

37, 33, 33, 32, 29, 28, 28, 23, 22, 22, 22, 21, 21, 21, 20, 20, 19, 19, 18, 18, 18, 18, 16, 15, 14, 14, 14, 12, 12, 9, 6

Use stem and leaf to display data

Solution: 

Numbers 3, 2, 1, and 0, arranged as a stems to the left of the bars. The other numbers come in the leaf part.

3|2337

2|001112223889

1|2244456888899

0|69

From this, data are concentrated in ones

Application activity 8.1.1

1. At the beginning of the school year, a Mathematics test was administered in year 1 to 50 student-teachers to test their level of understanding. Their results out of 20 were recorded as follows: 

Construct a frequency distribution table to help the tutor and the school administration to easily recognize the level of understanding for the Year 1 student-teachers. 

2. Differentiate qualitative and quantitative data from the list below: Product rating, basketball team classification, number of student-teachers in the classroom, weight, age, Number of rooms in a house, number of tutors in school.

8.1.2. Collection and presentation of grouped data

Activity 8.1.2

The mass of 50 tomatoes (measured to the nearest g) were measured and recorded in the table below

Construct a frequency distribution table, using equal class intervals of width 5g and taking the lower class boundary of the first interval as 84.5g.

CONTENT SUMMARY

When the range of data is large, the data must be grouped into classes that are more than one unit in width. In this case a grouped frequency distribution is used. Data in this case are grouped in a frequency distribution using groups or classes.

• Class limits: The class limits are the lower and upper values of the class

• Lower class limit: Lower class limit represents the smallest data value that can be included in the class.

• Upper class limit: Upper class limit represents the largest data value that can be included in the class.

• Class boundaries: Class boundaries are the midpoints between the upper class limit of a class and the lower class limit of the next class. Therefore, each class has a lower and an upper class boundary.

Example: The following data represent the marks obtained by 40 students in Mathematics test. Organize the data in the frequency table; grouping the values into classes, starting from 41-50

54 83 67 71 80 65 70 73 45 60 72 82 79 78 65 54 67 64 54 76 45 63 49 52

60 70 81 67 45 58 69 53 65 43 55 68 49 61 75 52

Solution:

Example

Using the frequency table above, determine the class boundaries of the three classes.

Solution 

For the first class, 41-50

The lower class boundary is the midpoint between 40 and 41, that is 40.5

The upper class boundary is the midpoint between 50 and 51, that is 50.5

For the second class, 51-60

The lower class boundary is the midpoint between 50 and 51, that is 50.5

The upper class boundary is the midpoint between 60 and 61, that is 60.5

For the third class, 61-70

The lower class boundary is the midpoint between 60 and 61, that is 60.5

The upper class boundary is the midpoint between 70 and 71, that is 70.5

Class width

Class width is the difference between the upper class boundary and lower class boundary

Application activity 8.1.2

Suppose a researcher wished to do a study on the distance (in kilometres) that the employees of a certain department store travelled to work each day. The researcher collected the data by asking each employee the approximate distance is the store from his or her home. Data are collected in original form called raw data as follow:

Since little information can be obtained from looking at raw data, help the researcher to make a frequency distribution table (use 1-3 km and 16-18 km as class limits) so that some general observations can easily be done by the researcher. 

8.2 Measure of central tendencies: Mean, median and mode

Activity 8.2

1. Observe the following marks of 10 students in mathematics test: 5, 4, 10, 3, 3, 4, 7, 4, 6, 5 and complete the table below

Calculate the mean, median, mode and range of this set

2. In three classes of Year three in TTC, during the quiz of mathematics out of 5 marks, 100 student-teachers obtain marks as shown in the table below:

• What is the marks obtained by most of the students?

• You are asked to calculate the mean mark for the class. Explain how you should find it. 

• What is the marks obtained by most of the students?

• You are asked to calculate the mean mark for the class. Explain how you should find it. 

CONTENT SUMMARY

For the comparison of one frequency distribution with another, it requires to summarize it in such a manner that its essence is expressed in few figures only. For this purpose, we find most representative value of data. This representative value of the data is known as the measure of central tendency or averages. 

a) Mean, mode , median and range of ungrouped data

Thus for any particular set of ungrouped data, it is possible to select some typical values or average to represent or describe the entire set such a descriptive value is referred to as a measure of central tendency or location such as mean, mode, median 

The median: 

The data is arranged in order from the smallest to the largest, the middle number is then selected. This really the central number of the range and is called the median.

Mean, mode, median and range of grouped data 

For any particular set of grouped data, it is possible to select some typical values or average to represent or describe the entire set such a descriptive value is referred to as a measure of central tendency or location such as mean, mode, median and range.

Range

In the case of grouped data the range is defined as the difference between the upper limit of the highest class and the lower limit of the smallest class.

Application activity 8.2

1. A group of student-teachers from SME were asked how many books they had read in previous year; the results are shown in the frequency table below. Calculate the mean, median and mode number of books read.

2. During oral presentation of internship report for year three studentteachers the first 10 student-teachers scored the following marks out of 10:

8, 7, 9, 10, 8, 9, 8, 6, 7 and 10

a) Calculate the mean of the group

b) Calculate the median and Mode

8.3 Graphical representation of grouped and ungrouped data

Activity 8.3

1. In a class of year 1, student-teachers are requested to provide their sizes in order to be given their sweaters. The graph below shows the sizes of sweaters for 30 students. Observe and interpret it by finding out the number of students with small, medium, large, extra-large. Guess the name of the graph.

2. The mass of 50 tomatoes (measured to the nearest g) were measured and recorded in the table below.

a) Determine cumulative frequency 

 b) Try to draw a histogram, the frequency polygon and the cumulative frequency polygon to illustrate the data.

3. A pie chart shows the mass (in tonnes) of each type of food eaten in a certain month. Observe the pie chart and make a corresponding frequency distribution table by considering the number of tones for each type of food as frequency. 

CONTENT SUMMARY

After the data have been organized into a frequency distribution, they can be presented in a graphical form. The purpose of graphs in statistics is to convey the data to the viewers in a pictorial form. It is easier for most people to comprehend the 

meaning of data presented graphically than data presented numerically in tables or frequency distributions. 

The most commonly used graphs are: Bar graph, Pie chart, Histogram, Frequency polygon, Cumulative frequency graph or Ogive. 

A bar graph 

A  bar  chart or  bar graph  is a chart or  graph  that presents numerical data with rectangular bars with heights or lengths proportional to the values that they represent. 

The bars can be plotted vertically or horizontally. A vertical bar chart is sometimes called a line graph

Pie chart 

 A pie chart is used to display a set of categorical data. It is a circle, which is divided into segments. Each segment represents a particular category. The area of each segment is proportional to the number of cases in that category.

Example: 

A TTC tutor conducted a survey to check the level of how student-teachers like some subjects (English, Mathematics, French, Entrepreneurship, Kinyarwanda) taught in SME option. The survey was done on 60 student-teachers and the table below summarizes the results.

In order to clearly present his/ her findings, the tutor presented the data on a bar graph and then on a pie chart as follows: 

1. Bar graph

Histogram

Histogram is a statistical graph showing frequency of data. The horizontal axis details the class boundaries, and the vertical axis represents the frequency. Blocks are drawn such that their areas (rather than their height, as in a bar chart) are proportional to the frequencies within a class or across several class boundaries. There are no spaces between blocks.

Example: Draw a histogram for the frequency distribution given in the table below.

Step 2: Draw Histogram (frequency against class boundaries)

Frequency polygon 

In a frequency polygon, a line is drawn by joining all the midpoints of the top of the bars of a histogram. 

A frequency polygon gives the idea about the shape of the data distribution. The two end points of a frequency polygon always lie on the x-axis.

Example 

Use the histogram above to draw a frequency polygon.

Solution

Example:

Construct a histogram, a frequency polygon, a cumulative frequency graph or ogive to represent the data shown below for the record high temperatures for each of the 50 states 

Application activity 8.3

1. At the beginning of the school year, a Mathematics test was administered in year 1 for 50 student-teachers to test their levels in this subject.. Their results out of 20 were recorded as follow:

Present the data using a bar graph to help the tutor and the school administration to easily recognize the level of understanding for the Year 1 student-teachers. 

2. The cumulative frequency distribution table below shows distances (in km) covered by 20 runners during the week of competition.

a) Construct a histogram

b) Construct a frequency polygon

c) Construct an ogive.

8.4 Measure of dispersion: Range, variance, Standard Deviation and coefficient of variation

Activity 8.4

CONTENT SUMMARY

The word dispersion has a technical meaning in statistics. The average measures the center of the data. It is one aspect of observations. Another feature of the observations is how the observations are spread about the center. The observation may be close to the center or they may be spread away from the center. If the observations are close to the center (usually the arithmetic mean or median), we say that dispersion, scatter or variation is small. If the observations are spread away from the center, we say that dispersion is large.

The study of dispersion is very important in statistical data. If in a certain factory there is consistence in the wages of workers, the workers will be satisfied. But if some workers have high wages and some have low wages, there will be unrest among the low paid workers and they might go on strikes and arrange demonstrations. If in a certain country some people are very poor and some are very high rich, we say there is economic disparity. It means that dispersion is large. 

The extent or degree in which data tend to spread around an average is also called the dispersion or variation. Measures of dispersion help us in studying the extent to which observations are scattered around the average or central value. Such measures are helpful in comparing two or more sets of data with regard to their variability.

Properties of a good measure of dispersion

i. It should be simple to calculate and easy to understand

ii. It should be rigidly defined

iii. Its computation be based on all the observations

iv. It should be amenable to further algebraic treatment

Some measures of dispersion are Quartiles, variance, Range, standard deviation, coefficient of variation

Variance

Variance measures how far a set of numbers is spread out. A variance of zero indicates that all the values are identical. Variance is always non-negative: a small variance indicates that the data points tend to be very close to the mean and hence to each other, while a high variance indicates that the data points are very spread out around the mean and from each other.

The variance is denoted and defined by

Example 1

Calculate the variance of the following distribution: 9, 3, 8, 8, 9, 8, 9, 18

STANDARD DEVIATION

The standard deviation has the same dimension as the data, and hence is comparable to deviations from the mean. We define the standard deviation to be the square root of the variance. thus, the standard deviation is denoted and defined by

Example 1

The six runners in a 200 meter race clocked times (in seconds) of 24.2, 23.7, 25.0, 23.7, 24.0,4.6

a) Find the mean and standard deviation of these times.

b) These readings were found to be 10% too low due to faulty timekeeping. 

Write down the new mean and standard deviation.

Example 2

The heights (in meters) of six children are 1.42, 1.35, 1.37, 1.50, 1.38 and 1.30. 

Calculate the mean height and the standard deviation of the heights

Coefficient of variation

The coefficient of variation measures variability in relation to the mean (or average) and is used to compare the relative dispersion in one type of data with the relative dispersion in another type of data. It allows us to compare the dispersions of two different distributions if their means are positive. The greater dispersion corresponds to the value of the coefficient of greater variation.

The coefficient of variation is a calculation built on other calculations: the standard deviation and the mean as follows: 

Example:

One data series has a mean of 140 and standard deviation 28.28. The second data series has a mean of 150 and standard deviation 24. Which of the two has a greater dispersion?

Solution:

Range

In the case of ungrouped data, the range of a set of observations is the difference in values between the largest and the smallest observations in the set. In the case of grouped data the range is defined as the difference between the upper limit of the highest class and the lower limit of the smallest class.

Example

Calculate the range of the following set of the data set: 1, 3, 4, 5, 5, 6, 9, 14 and 21

Solution: From the given series the lowest data is 1 and the highest data is 21

The Range = highest value −lowest value

Range = −= 21 1 20

Application activity 8.4

1. Out of 4 observations done by tutor of English, arranged in descending order, the 5th, 7th, 8th and 10th observations are respectively 89, 64, 60 and 49. 

Calculate the median of all the 4 observations.

2. In the following statistical series, calculate the standard deviation of the following set of data 56,54,55,59,58,57,55

3. In the classroom of SME the first five student-teachers scored the following marks out of 10 in a quiz of Mathematics

 5, 6, 5, 2, 4, 7, 8, 9, 7, 5

a) Calculate the mean, median and the modal mark

b) Determine the quartiles and interquartile range

c) Calculate the variance and the standard deviation

d) Determine the coefficient of variation 

8.5 Application of statistics in daily life

Activity 8.5

Using internet or reference books from the school library, make a research to provide in written form at least one example of where the following statistical terminologies are needed and used in real life situations: 

• Frequency distribution 

• Statistical graphs like Bar graph, Histogram, Frequency polygon Mean, 

• Variance 

Make a presentation of your findings to the whole class 

CONTENT SUMMARY

Statistics is concerned with scientific method for collecting and presenting, organizing and summarizing and analyzing data as well as deriving valid conclusions and making reasonable decisions on the basis of this analysis 

Today, statistics is widely employed in government, business and natural and social sciences. Statistical methods are applied in all field that involve decision making. 

For example, in agriculture, statistics can help to ensure the amount of crops are grown this year in comparison to previous year or in comparison to required amount of crop for the country. It may also be helpful to analyze the quantity and size of grains grown due to use of different fertilizer. 

In education, statistics can be used to analyze and decide on the money spend on girls education in comparison to boys education.

 Nowadays, graphs are used almost everywhere. In stock market, graphs are used to determine the profit margins of stock. There is always a graph showing how prices have changed over time, food price, budget forecasts, exchange rates…

Mean is used as one of comparing properties of statistics. It is defined as the average of all the clarifications. Mean helps Teachers to see the average marks of the students. 

A large standard deviation indicates that the data points can spread far from the mean and a small standard deviation indicates that they are clustered closely around the mean. Standard deviation is often used to compare real-world data against a model to test the model

In finance, standard deviation is often used as a measure of the risk associated with price-fluctuations of a given asset (stocks, bonds, property, etc.), or the risk of a portfolio of assets, Standard deviation provides a quantified estimate of the uncertainty of future returns.

Application activity 8.5

1. Using internet or reference books from the school library, make a research to provide in written form at least one example of where the following statistical terminologies are needed and used in real life situations: 

• Frequency distribution 

• Statistical graphs like Pie chart, cumulative frequency polygon or ogive 

• Mode and median

• Standard deviation 

Make a presentation of your findings to the whole class 

2. Collect data on 20 students’ heights in your class and organize them in a frequency distribution. Calculate the mean of 20 students’ heights, variance as well as standard deviation. What conclusion can you make based on the calculated mean, variance and standard deviation?

8.6. END UNIT ASSESSMENT 8

1. Use the graph below to answer the questions that follows 

a) Use the graph to estimate the mode. 

b) State the range of the distribution.

c) Draw a frequency distribution table from the graph

2. In test of mathematics, 10 student-teachers got the following marks:

6, 7, 8, 5, 7, 6, 6, 9, 4, 8

a) Calculate the mean, mode, quartiles and interquartile range

b) Calculate the variance and standard deviation

c) Calculate the coefficient of variation

3. A survey taken in a restaurant shows the following number of cups of coffee consumed with each meal. Construct an ungrouped frequency distribution. 0 2 2 1 1 2 3 5 3 2 2 2 1 0 1 2 4 4 0 1 0 1 4 4 2 2 0 1 1 5

4. The amount of protein (in grams) for a variety of fast-food sandwiches is reported here. Construct a frequency distribution using six classes. Draw a histogram, frequency polygon, and ogive for the data, using relative frequencies. Describe the shape of the histogram. 

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