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UNIT 1: ARITHMETICS
Key Unit competence: Use arithmetic operations to solve simple real life problems
1.O. Introductory Activity
The simple interest earned on an investment is I prt = where I is the
interest earned, p is the principal, r, is the interest rate and t is the time in
years. Assume that 50,000Frw is invested at annual interest rate of 8%
and that the interest is added to the principal at the end of each year.
a. Discuss the amount of interest that will be earned each year for 5
years.
b. How can you find the total amount of money earned at the end of
these 5 years? Classify and explain all Mathematics operations
that can be used to find that money.1.1 Fractions and related problems
Activity 1.1
1) Sam had 120 teddy bears in his toy store. He sold 2/3 of them at 12
Rwandan francs each. How much did he receive?
2) Simplify the following fractions and explain the method used to simplify.
3) Given that the denominator is different from zero,
4) Do you some times use fractions in your life? Explain your answer.
CONTENT SUMMARY
Consider the expressions
in each of these, the numerator or
the denominator or both contain a variable or variables. These are examples
of algebraic fractions. Since the letter used in these fractions stand, for real
numbers, we deal with algebraic fractions in the same way as we do with
fractions in arithmetic. Fractions can be simplified and operations on fractions
such as addition, subtraction, multiplication and division can be applied in
simple arithmetic for solving related problems.Adding fractions:
To add fractions there is a simple rule
Dividing fractions
To divide fractions, first flip the fraction we want to divide by, then use the
same method as for multiplication
A method of calculating the interest rate on an investment is explained in the
UNIT2 : EQUATIONS AND INEQUALITIES
Key Unit competence: Model and solve daily life problems using
linear, quadratic equations or inequalities
2.O. Introductory Activity
1) By the use of library and computer lab, do the research and explain
the linear equation.
2) If x is the number of pens for a learner, the teacher decides to give
him/her two more pens. What is the number of pens will have a learner
with one pen?
a) Complete the following table called table of value to indicate the number
of pens for a learner who had x pens for x ≥ 0 .
CONTENT SUMMARY
An equation is a mathematical statement expressing the equality of two
quantities or expressions. Equations are used in every field that uses real
numbers. A linear equation is an equation of a straight line.
Real life problems involving linear equations
Problems which are expressed in words are known as problems or applied
problems. A word or applied problem involving unknown number or quantity
can be translated into linear equation consisting of one unknown number
or quantity. The equation is formed by using conditions of the problem. By
solving the resulting equation, the unknown quantity can be found.
In solving problem by using linear equation in one unknown the following
steps can be used:
i) Read the statement of the word problems
ii) Represent the unknown quantity by a variable
iii) Use conditions given in the problem to form an equation in the
unknown variable
iv) Verify if the value of the unknown variable satisfies the conditions of
the problem.
Examples
1) The sum of two numbers is 80. The greater number exceeds the smaller
number by twice the smaller number. Find the numbers.
2.6 Solving algebraically and graphically simultaneous linear inequalities in two unknown
2.7 Solving quadratic equations by the use of factorization and discriminant
Activity 2.7
Smoke jumpers are fire fighters who parachute into areas near forest
fires. Jumpers are in free fall from the time they jump from a plane until
they open their parachutes. The function
gives a jumper’s height y in metre after t seconds for a jump from1600m.
a) How long is free fall if the parachute opens at1000m?
b) Complete a table of values for t = 0, 1, 2, 3, 4, 5 and 6.
CONTENT SUMMARY
CONTENT SUMMARY
When only two or single variables and equations are involved, a simultaneous equation system can be related to familiar graphical solutions, such as supply and demand analysis.
2.9. END UNIT ASSESSMENT
UNIT 3: DESCRIPTIVE STATISTICS
3.0. Introductory Activity
1) 1. At the market a fruit-seller has the following daily sales Rwandan francs for five consecutive days: 1000Frw, 1200Frw, 125Frw, 1000Frw, and 1300Frw. Help her to determine the money she could get if the sales are equally distributed per day to get the same total amount of money.
2) During the welcome test of Mathematics for the first term 10 studentteachers of year one language education scored the following marks
out of 10 : 3, 5,6,3,8,7,8,4,8 and 6.
a) What is the mean mark of the class?
b) Chose the mark that was obtained by many students.
c) Compare the differences between the mean of the group and the
mark for every student teacher.
3.1 Definition and type of data
Learning activities 3.1
Carry out research on statistics to determine the meanings of statistics
and types of data. Use your findings to select qualitative and quantitative
data from this list: Male, female, tall, age, 20 sticks, 45 student-teachers,
and 20 meters, 4 pieces of chalk.
CONTENT SUMMARY
Statistics is the branch of mathematics that deals with data collection, data
organization, summarization, analysis and draws conclusions from data.
The use of graphs, charts, and tables and the calculation of various
statistical measures to organize and summarize information is called
descriptive statistics. Descriptive statistics helps to reduce our information to
a manageable size and put it into focus.
Every day, we come across a wide variety of information in form of facts,
numerical figures or table groups. A variable is a characteristic or attribute
that can assume different values. Data are the values (measurements or
observations) that the variables can assume.
Variables whose values are determined by chance are called random
variables. A collection of data values forms a data set. Each value in the
data set is called a data value or a datum.
For example information related to profit/ loss of the school, attendance
of students and tutors, used materials, school expenditure in term or year,
etc. These facts or figure which is numerical or otherwise, collected with
a definite purpose is called data. This is the word derived from Latin word
Datum which means pieces of information.
Qualitative variable
The qualitative variables are variables that cannot be expressed using
a number. A qualitative data is determined when the description of the
characteristic of interest results is a non-numerical value. A qualitative
variable may be classified into two or more categories. Data obtained by
observing values of a qualitative variable are referred to as qualitative data.
The possible categories for qualitative variables are often coded for the
purpose of performing computerized statistical analysis so that we have
qualitative data.
Quantitative variable
Quantitative variables are variables that are expressed in numerical terms,
counted or compared on a scale. A quantitative data is determined when the
description of the characteristic of interest results in numerical value. When
measurement is required to describe the characteristic of interest or when it
is necessary to perform a count to describe the characteristic, a quantitative
variable is defined.
Discrete data is a quantitative data whose values are countable. Discrete
data usually result from counting.
Continuous data is a quantitative data that can assume any numerical
value over an interval or over several intervals. Continuous data usually
results from making a measurement of some type. Data obtained by
observing values of a quantitative variable are referred to as quantitative
data. Quantitative data obtained from a discrete variable are also referred to
as discrete data and quantitative data obtained from a continuous variable
are called continuous data.
Data can be used in different ways. The body of knowledge called statistics
is sometimes divided into two main areas, depending on how data are used.
The two areas are descriptive statistics and inferential statistics.
Descriptive statistics consists of the collection, organization, summarization,
and presentation of data.
Inferential statistics consists of generalizing from samples to populations,
performing estimations and hypothesis tests, determining relationships
among variables, and making predictions.
A population consists of all subjects (human or otherwise) that are being
studied.
A sample is a group of subjects selected from a population.
3.2 Data presentation or organization
Activity 3.2
1) At the beginning of the school year student-teachers came to school
with all requirements materials and others don’t attend the school on
first day. The table below shows the number of student-teachers who
attended the school in 5 classrooms on first day.
Present the above data on bar chart and discuss what you can say when interpreting the chart.
2) One tutor of TTC wants to check the level of how his/her student-teachers like different subjects taught in Language Education option. The survey is done on 60 student-teachers. Here is the number of participants of the survey.
CONTENT SUMMARY
After the collection of data, the researcher needs to organize and present
them in order to help those who will benefit from the research and lead him
or her to the conclusion. When the data are in original form, they are called
raw data and are listed next.
Raw data
After the collection of data, one can present them in the raw form presentation.
Example
The following are numbers of notebooks for 54 student-teachers.
3 4 5 6 6 7 3 2 5 4 5 7 4 3 2 3 6 5
8 5 6 4 2 6 5 3 4 7 9 8 6 7 4 5 4 6
3 4 3 6 7 8 2 7 6 5 4 6 4 7 8 9 9 5
Since little information can be obtained from looking at raw data, the
researcher organizes the data into what is called a frequency distribution. A
frequency distribution consists of the number of times each values appears
in the raw data and sometimes the corresponding percentage vis-à-vis the
total number in the sample.
1. Frequency distribution
Data can be presented in various forms depending on the type of data
collected. A frequency distribution is the organization of raw data in the
table form by the use of frequencies.
Example
The following data represent marks obtained by 12 student-teachers out of
20 in mathematics test of a certain TTC.
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:
The total frequency is the number of items in the population. In the survey
about the marks of student-teachers above, the total frequency is the number
of all student-teachers. We find this by adding up the numbers from the third
column of the table: 2112 51 +++ + + which is equal to 12 .
Grouped data
When the number of data is too large, a simple distribution is not appropriate. In this case we come up with a grouped frequency distribution. A grouped frequency distribution organizes data into groups or classes.
Definitions related to grouped frequency distribution
a) Class limits: The class limits are the lower and upper values of the class
b) Lower class limit: Lower class limit represents the smallest data value that can be included in the class.
c) Upper class limit: Upper class limit represents the largest data value that can be included in the class.
d) Class mark or class midpoint:
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, stating 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 v69 53 65 43 55 68 49 61 75 52.
The classes: 41-50, 51-60, 71-80, 81-90
Lower class limits: 41, 51, 61, 71, and 81
Upper class limits: 50, 60, 70, 80, and 90
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.
For [10,15[ the lower class boundary is 10, The upper class boundary is 15 Class width =−= 15 10 5
We have Categorical Frequency Distributions and Grouped Frequency Distributions.
The categorical frequency distribution is used for data that can be placed in specific categories, such as nominal- or ordinal-level data.
Example:
Data such as political affiliation, religious affiliation, or major field of study would use categorical frequency distributions.
Example:
Twenty-five army inductees were given a blood test to determine their blood type.
The data set is:
Grouped Frequency Distributions
When the range of the data is large, the data must be grouped into classes
that are more than one unit in width, in what is called a grouped frequency
distribution. For example, a distribution of the number of hours that boat
batteries lasted is the following:
The procedure for constructing a grouped frequency distribution, i.e., when
the classes contain more than one data value
Example
These data represent the record high temperatures in degrees Fahrenheit
(*F) for each of the 50 states. Construct a grouped frequency distribution for
the data using 7 classes.
1) Determine the classes
– Find the highest value and lowest value: H =134 and L=100.
– Find the range: R = highest value-lowest value H -L, so
R=134 -100 = 34
– Select the number of classes desired (usually between 5 and 20). In this case, 7 is arbitrarily chosen.
– Find the class width by dividing the range by the number of classes
– Select a starting point for the lowest class limit. This can be the smallest
data value or any convenient number less than the smallest data value.
In this case, 100 is used. Add the width to the lowest score taken as the
starting point to get the lower limit of the next class. Keep adding until
there are 7 classes, as shown, 100, 105, 110, etc.
– Subtract one unit from the lower limit of the second class to get the
upper limit of the first class. Then add the width to each upper limit to
get all the upper limits. 105-1=104.
The first class is 100–104, the second class is 105–109, etc.
– Find the class boundaries by subtracting 0.5 from each lower class
limit and adding 0.5 to each upper class limit:
99.5–104.5, 104.5–109.5, etc.
2) Tally the data
3) Find the numerical frequencies from the tallies.
The completed frequency distribution is
2. 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.
4. Histograms, Frequency Polygons, and Ogives
After you have organized the data into a frequency distribution, you can present them in graphical form. The purpose of graphs in statistics is to convey the data to the viewers in 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 three most commonly used graphs in research are
1) The histogram.
2) The frequency polygon.
3) The cumulative frequency graph or ogive (pronounced o-jive).
a) The Histogram
The histogram is a graph that displays the data by using contiguous vertical bars (unless the frequency of a class is 0) of various heights to represent the frequencies of the classes.
Example:
Construct a histogram to represent the data shown for the record high temperatures for each of the 50 states.
Step 1: Draw and label the x and y axes. The x axis is always the horizontal axis, and the y axis is always the vertical axis.
Step 2: Represent the frequency on the y axis and the class boundaries on the x axis.
Step 3: Using the frequencies as the heights, draw vertical bars for each class. See Figure below
b)The Frequency Polygon
The frequency polygon is a graph that displays the data by using lines that
connect points plotted for the frequencies at the midpoints of the classes.
The frequencies are represented by the heights of the points.
Example:
Using the frequency distribution given in Example 2–4, construct a frequency polygon
Step 1 Find the midpoints of each class. Recall that midpoints are found by
adding the upper and lower boundaries and dividing by 2:
Step 2 Draw the x and y axes. Label the x axis with the midpoint of each class, and then use a suitable scale on the y axis for the frequencies.
Step 3 Using the midpoints for the x values and the frequencies as the y values, plot the points.
Step 4 Connect adjacent points with line segments. Draw a line back to the x axis at the beginning and end of the graph, at the same distance that the previous and next midpoints would be located, as shown in Figure 2–3.
The frequency polygon and the histogram are two different ways to represent
the same data set. The choice of which one to use is left to the discretion of
the researcher.
c) The Ogive
The ogive is a graph that represents the cumulative frequencies for the
classes in a frequency distribution.
Step 1: Find the cumulative frequency for each class.
Step 2: Draw the x and y axes. Label the x axis with the class boundaries. Use an appropriate scale for the y axis to represent the cumulative frequencies.
(Depending on the numbers in the cumulative frequency columns, scales such as 0, 1, 2, 3, . . . , or 5, 10, 15, 20, . . . , or 1000, 2000, 3000, . . . can be used.
Do not label the y axis with the numbers in the cumulative frequency column.)
In this example, a scale of 0, 5, 10, 15, . . . will be used.
Step 3 Plot the cumulative frequency at each upper class boundary, as shown in Figure below. Upper boundaries are used since the cumulative frequencies represent the number of data values accumulated up to the upper boundary of each class.
Step 4 Starting with the first upper class boundary, 104.5, connect adjacent points with line segments, as shown in the figure. Then extend the graph to the first lower class boundary, 99.5, on the x axis.
Cumulative frequency graphs are used to visually represent how many values are below a certain upper class boundary. For example, to find out how many record high temperatures are less than 114.5_F, locate 114.5_F on the x axis, draw a vertical line up until it intersects the graph, and then draw a horizontal line at that point to the y axis. The y axis value is 28, as shown in the figure.
5. Relative Frequency Graphs
The histogram, the frequency polygon, and the ogive shown previously were constructed by using frequencies in terms of the raw data. These distributions can be converted to distributions using proportions instead of raw data as frequencies. These types of graphs are called relative frequency graphs.
Example:
Construct a histogram, frequency polygon, and ogive using relative frequencies for the distribution (shown here) of the kilometers that 20 randomly selected runners ran during a given week.
Solution:
Step 1: Convert each frequency to a proportion or relative frequency by dividing the frequency for each class by the total number of observations.
Place these values in the column labeled Relative frequency.
Step 2: Find the cumulative relative frequencies. To do this, add the frequency in each class to the total frequency of the preceding class. In this case, 0 +0.05 = 0.05, 0.05 + 0.10 = 0.15, 0.15 +0.15 =0.30,
0.30+0.25=0.55, etc. Place these values in the column labeled Cumulative relative frequency.
An alternative method would be to find the cumulative frequencies and then convert each one to a relative frequency.
Step 3: Draw each graph as shown in Figure 2–7. For the histogram and ogive, use the class boundaries along the x axis. For the frequency polygon, use the midpoints on the x axis. The scale on the y axis uses proportions.
6. Pie chart
A pie chart is used to display a set of categorical data. It is a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution.
Since there are 0 360 in a circle, the frequency for each class must be converted into a proportional part of the circle. This conversion is done by using the formula.
where f is frequency for each class and n is the sum of the frequencies.
Hence, the following conversions are obtained. The degrees should sum to
360.
Each frequency must also be converted to a percentage by using the formula
Next, using a protractor and a compass, draw the graph using the appropriate
degree measures found in step 1, and label each section with the name and
percentages.
Example:
1) One tutor in TTC want to check the level of how his/her student and teachers like different subject taught in Language Education option. The survey done on 60 student-teachers in English, Mathematics, French, Entrepreneurship and Kinyarwanda.
Here is the results he/she obtained respectively after making survey 12, 24, 6, 10, and 8. Present the data on pie chart.
2) Construct a pie graph showing the blood types of the army inductees described above. The frequency distribution is repeated here.
Solution:
Step 1: Find the number of degrees for each class, using the formula.
Step 3: Using a protractor, graph each section and write its name and corresponding percentage, as shown in the Figure
7. Stem and Leaf Plots
The stem and leaf plot is a method of organizing data and is a combination
of sorting and graphing. It uses part of the data value as the stem and part
of the data value as the leaf to form groups or classes. It has the advantage
over a grouped frequency distribution of retaining the actual data while
showing them in graphical form.
Examples:
1) At an outpatient testing center, the number of cardiograms performed
each day for 20 days is shown. Construct a stem and leaf plot for the data.
Step 1: Arrange the data in order: 02, 13, 14, 20, 23, 25, 31, 32, 32, 32,
32, 33, 36, 43, 44, 44, 45, 51, 52, 57
Note: Arranging the data in order is not essential and can be cumbersome
when the data set is large; however, it is helpful in constructing a stem
and leaf plot. The leaves in the final stem and leaf plot should be arranged
in order.
Step 2 Separate the data according to the first digit, as shown.
02 13, 14 20, 23, 25 31, 32, 32, 32, 32, 33, 36, 43, 44, 44, 45 51, 52, 57
Step 3: A display can be made by using the leading digit as the stem and
the trailing digit as the leaf.
For example, for the value 32, the leading digit, 3, is the stem and the
trailing digit, 2, is the leaf. For the value 14, the 1 is the stem and the 4 is
the leaf. Now a plot can be constructed as follows:
It shows that the distribution peaks in the center and that there are no
gaps in the data. For 7 of the 20 days, the number of patients receiving
cardiograms was between 31 and 36. The plot also shows that the testing
center treated from a minimum of 2 patients to a maximum of 57 patients
in any one day.
If there are no data values in a class, you should write the stem number
and leave the leaf row blank. Do not put a zero in the leaf row.
2) 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 using 2, 3,
and 4 as your stems.
3) 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.
Application activity 3.2
1) Suppose that a tutor conducted a test for student-teachers and the
marks out of 10 were as follows: 3 3 3 5 6 4 6 7 8 3 8 8 8 10 9
10 9 10 8 10 6
a) Draw a frequency table;
b) Draw a relative frequency table and calculate percentage for each;
c) Present data in cumulative frequency table, hence show the
number of student-teachers who did the test.
2) During the examination of English, student-teachers got the following
results out of 80: 54, 42, 61, 47, 24, 43, 55, 62, 30, 27, 28, 43, 54, 46,
25, 32, 49, 73, 50, 45.
Present the results using stem and leaf.
3) A firm making artificial sand sold its products in four cities: 5% was
sold in Huye, 15% in Musanze, 15% in Kayonza and 65% was sold
in Rwamagana.
a) What would be the angles on pie chart?
b) Draw a pie chart to represent this information.
c) Use the pie chart to comment on these findings.
3.3 Graph interpretation and Interpretation of statistical data
Activity 3.3
The graph below shows the sizes of sweaters worn by 30 year 1 students
in a certain school. Observe it and interpret it by answering the questions
below it:
CONTENT SUMMARY
Once data has been collected, they may be presented or displayed in various ways including graphs. Such displays make it easier to interpret and compare the data.
Examples
1) The bar graph shows the number of athletes who represented five African countries in an international championship.
a) What was the total number of athletes representing the five countries?
b) What was the smallest number of athletes representing one country?
c) What was the most number of athletes representing a country?
d) Represent the information on the graph on a frequency table.
Solution:
We read the data on the graph:
a) Total number of athletes are: 18 + 10 + 22 + 6 + 16 = 72 athletes
b) 6 athletes
c) 22 athletes
d) Representation of the given information on the graph on a frequency table
2) Use a scale vertical scale 2cm: 10 students and Horizontal scale 2cm: 10
represented on histogram below to answers the questions that follows
a) Estimate the mode
b) Calculate the range
Solution:
a) To estimate the mode graphically, we identify the bar that represents
the highest frequency. The mass with the highest frequency is 60 kg.
It represents the mode.
b) The highest mass = 67 kg and the lowest mass = 57 kg
Then, The range=highest mass-lowest mass=67 57 10 kg kg kg
3.4 Measures of central tendencies for ungrouped data
Activity 3.4
Conduct a research in the library or on the internet and explain measures
of central tendency, their types and provide examples.
Insist on explaining how to determine the Mean, Mode, Median and their role
when interpreting statistic data.
CONTENT SUMMARY
Measures of central tendency were studies in S1 and S2.
1. The mean
The mean, also known as the arithmetic average, is found by adding the values of the data and dividing by the total number of values. Suppose that a fruit seller earned the flowing money from Monday to Friday respectively: 300, 200, 600, 500, and 400 Rwandan francs. The mean of this money explains the same daily amount of money that she should earn to totalize the same amount in 5 days.
Where:
L: the lower limit of the modal class
fm: the modal frequency
f1: the frequency of the immediate class below the modal class
f2: the frequency of the immediate class above the modal class
w: modal class width.
Example:
Find the modal class for the frequency distribution of kilometers that 20
runners ran in one week.
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 observation 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.
1. Quartile
A measure which divides any array into four equal parts is known as quartile.
Each proportion contains equal number of items. The fist and the second
and the third points are termed as first quartile. The first quartile has 25% of
the items of the distribution below it and 75% of the items are greater than it.
The second quartile which is the median has 50% of the observations above
it and 50% of the observations below it. For the arranged data in ascending
order and quartiles are calculated as follows:
With observation, data are collected through direct observation. Information
could also be collected using an existing table that shows types of data to
collect in a certain period of time, this method of collecting data is reliable
and accurate.
Once data has been collected, they may be presented or displayed in various
ways. Such displays make it easier to interpret and compare the data.
After the collection of data there is need of interpreting them and here there
are some tips:
– Collect your data and make it as clean as possible.
– Choose the type of analysis to perform: qualitative or quantitative
– Analyze the data through various statistical methods such as mean,
mode, standard deviation or Frequency distribution tables
– Reflect on your own thinking and reasoning and analyze your data and
then interpret them referring to the reality.
During interpretation, avoid subjective bias, false information and inaccurate
decisions.
