◄Unit 4 Univariate Statistics and ApplicationsTopic 6►

Unit 5 Bivariate statistics and Applications

  • Unit 5 Bivariate statistics and Applications

    Key unit competence: Apply bivariate statistical concepts to collect, organise,
    analyse, present, and interpret data to draw appropriate

    decisions.

    Introductory activity

    In Kabeza village, after her 9 observations about farming, UMULISA saw
    that in every house observed, where there is a cow (X) if there is also

    domestic duck (Y), then she got the following results:

    (1,4) , (2,8) , (3,4) , (4,12) , (5,10)

    (6,14) , (7,16) , (8,6) , (9,18)

    d

    a) Represent this information graphically in (x, y) − coordinates .
    b) Find the equation of line joining any two point of the graph and
    guess the name of this line.
    c) According to your observation from (a), explain in your own
    words if there is any relationship between Cows (X) and

    domestic duck (Y).

    5.1 Introduction to bivariate statistics.

    5.1.1. Key concepts of bivariate statistics

    Learning Activity 5.1.1

    d

    If a businessman wants to make future investments based on how ice cream

    sales relate to the temperature,

    i) Which statistical measure and variables will he use?

    ii) Which variable depends on the other?

    iii) In statistics, how do we call a variable which depends on the other?

    iv) Plot the corresponding points (x, y) on a Cartesian plane and describe
    the resulting graph. Can you draw a line roughly representing the

    points in your graph?

    v) Can you obtain a rule relating x and y ? Explain your answer.

    CONTENT SUMMARY

    Bivariate data is data that has been collected in two variables, and each data
    point in one variable has a corresponding data point in the other value. Bivariate
    data is observation on two variables, whilst univariate data is an observation on
    only one variable. We normally collect bivariate data to try and investigate the
    relationship between the two variable sand then use this relationship to inform

    future decisions.

    Bivariate statistics deals with the collection, organization, analysis,
    interpretation, and drawing of conclusions from bivariate data. Data sets that
    contain two variables, such as wage and gender, and consumer price index and
    inflation rate data are said to be bivariate. In the case of bivariate or multivariate
    data sets we are often interested in whether elements that have high values of

    one of the variables also have high values of other variables.

    In bivariate statistics, we have independent variable and dependent variable.
    Dependent variable refers to variable that depends on the other variable (s).

    Independent variable refers to variable that affects the other variable (s).

    Scatter diagram is the graph that represents the bivariate data in x and y
    cartesian plane. The points do not lie on a line or a curve, hence the name. A
    scatter graph of bivariate data is a two-dimensional graph with one variable on
    one axis, and the other variable on the other axis. We then plot the corresponding
    points on the graph. We can then draw a regression line (also known as a line of
    best fit), and look at the correlation of the data (which direction the data goes,
    and how close to the line of best fit the data points are).For example, the scatter
    diagram showing the relationship between secondary school graduate rate and

    the % of residents who live below the poverty line.

    s

    Examples of bivariate statistics

    • Collecting the monthly savings and number of family members’ data
    of every family that constitutes your population if you are interested in
    finding the relationship between savings and number of family members.
    In this case, you will take a small sample of families from across the country
    to represent the larger population of Rwanda. You will use this sample to
    collect data on family monthly savings and number of family members.
    • We can collect data of outside temperature versus ice cream sales, these
    would both be examples of bivariate data. If there is a relationship showing
    an increase of outside temperature increased ice cream sales, then shops
    could use this information to buy more ice cream for hotter spells during

    the summer.

    Application activity 5.1.1

    1. Using an example, differentiate univariate statistics from bivariate
    statistics.
    2. Using an example, differentiate dependent variable from

    independent variable.

    5.2 Measures of linear relationship between two variables:
    covariance, Correlation, regression line and analysis,

    and spearman’s coefficient of correlation.

    5.2.1 Covariance and correlation.

    Learning Activity 5.2.1

    As students of economics, you might be interested in whether people with
    more years of schooling earn higher incomes. Suppose you obtain the data
    from one district for the population of all that district households. The data
    contain two variables, household income (measured in FRW) and a number
    of years of education of the head of each household.
    i) Which statistical measure will you use to know whether people with
    more years of schooling earn higher incomes.
    ii) If you want to know how household income and year of schooling

    covariate, which statistical measure will you consider?

    CONTENT SUMMARY

    Covariance is a statistical measure that describes the relationship between a
    pair of random variables where change in one variable causes change in another
    variable. It takes any value between -infinity to + infinity, where the negative
    value represents the negative relationship whereas a positive value represents
    the positive relationship. It is used for the linear relationship between variables.

    It gives the direction of relationship between variables.

    d

    The covariance of annual household income and schooling years from the

    learning activity 5.2.1 is given by

    f

    The covariance of variables x and y is a measure of how these two variables
    change together. If the greater values of one variable mainly correspond with the
    greater values of the other variable, and the same holds for the smaller values,
    i.e., the variables tend to show similar behavior, the covariance is positive. In
    the opposite case, when the greater values of one variable mainly correspond
    to the smaller values of the other, i.e., the variables tend to show opposite
    behavior, the covariance is negative. If covariance is zero, the variables are said
    to be uncorrelated, meaning that there is no linear relationship between them.

    Correlation

    The Pearson’s coefficient of correlation (or product moment coefficient of
    correlation or simply coefficient of correlation), denoted by r , is a measure

    of the strength of linear relationship between two variables. The coefficient

    w

    There are three types of correlation:

    • Negative correlation

    • Zero correlation

    • Positive correlation

    If the linear coefficient of correlation takes values closer to -1, the correlation is
    strong and negative, and will become stronger the closer r approaches -1.
    If the linear coefficient of correlation takes values close to 1, the correlation is
    strong and positive, and will become stronger the closer r approaches 1. If the
    linear coefficient of correlation takes values close to 0, the correlation is weak.
    If r = 1 or r=-1, there is perfect correlation and the line on the scatter plot is
    increasing or decreasing respectively. If r = 0, there is no linear correlation.

    f

    Application activity 5.1.1

    The production manager had ten newly recruited workers under him. For
    one week, he kept a record of the number of times that each employee
    needed help with a task and make a scatter diagram for the data. What

    type of correlation is there?

    s

    5.2.2 Regression line and analysis

    Learning Activity 5.2.2

    In a village, Emmanuella visited nine families and their farming
    activities. For each visited family there were x number of cows,
    and y goats. Emmanuella recorded her observations as follows:
    (1, 4),(2,8),(3, 4),(4,12),(5,10),(6,14),(7,16),(8,6),(9,18) .
    a) Represent Emmanuella’s recorded observations graphically in
    cartesian plane.
    b) Connect any two points on the graph drawn above to form a
    straight line and find equation of that line. How are the positions
    of the non-connected points vis-à-vis that line?
    c) According to your observation from a., is there any relationship
    between the variation of the number of cows and the number of
    goats? Explain.

    CONTENT SUMMARY

    Bivariate statistics can help in prediction of a value for one variable if we know
    the value of the other. We use the regression line to predict a value of y for any
    given value of x and vice versa. The “best” line would make the best predictions:
    the observed y-values should stray as little as possible from the line. This straight
    line is the regression line from which we can adjust its algebraic expressions

    and it is written as y = ax + b .

    Deriving regression line equation

    The regression line y on x has the form y = ax + b . We need the distance from
    this line to each point of the given data to be small, so that the sum of the square

    of such distances be very small.

    s

    1. Differentiate relation (1) with respect to b . In this case, x, y and a will

    be considered as constants.

    2. Equate relation obtained in 1 to zero, divide each side by n and give the

    value of b .

    • Take the value of b obtained in 2 and put it in relation obtained in 1.

    Differentiate the obtained relation with respect to Variance for variable x

    s

    s

    d

    c

    Application activity 5.1.1

    Consider the following data:

    d

    Find the regression line of y on x and deduce the approximated value of y

    when x=64.

    5.2.3 Spearman’s coefficient of correlation

    Learning Activity 5.2.2

    The death rate data from 1995 to 2000 for developed and underdeveloped

    countries are displayed in the table below.

    d

    i) Write the death rate for developed countries and the death rate for
    underdeveloped countries in ascending order.
    ii) Rank the death rate for developed countries and the death rate for
    underdeveloped countries such that the lowest death rate is ranked 1

    and the highest death rate is ranked 6.

    iii) Complete the table below using the information above.

    d

    d

    d

    Application activity 5.2.3

    Calculate Spearman’s coefficient of rank correlation for the series.

    s

    5.2.4 Application of bivariate statistics

    Learning Activity 5.2.4

    Referring to what you have learnt in this unit, discuss how bivariate

    statistics is used in our daily life.

    CONTENT SUMMARY

    Bivariate statistics can help in prediction of the value for one variable if we know
    the value of the other. Bivariate data occur all the time in real-world situations
    and we typically use the following methods to analyze the bivariate data:
    • Scatter plots
    • Correlation Coefficients
    • Simple Linear Regression
    The following examples show different scenarios where bivariate data appears
    in real life.
    Businesses often collect bivariate data about total money spent on advertising
    and total revenue. For example, a business may collect the following data for 12

    consecutive sales quarters:

    s

    This is an example of bivariate data because it contains information on exactly
    two variables: advertising spend and total revenue. The business may decide to
    fit a simple linear regression model to this dataset and find the following fitted
    model:
    Total Revenue = 14,942.75 + 2.70*(Advertising Spend). This tells the business
    that for each additional dollar spent on advertising, total revenue increases by
    an average of $2.70.
    • Economists often collect bivariate data to understand the relationship
    between two socioeconomic variables. For example, an economist may
    collect data on the total years of schooling and total annual income among

    individuals in a certain city:

    s

    He may then decide to fit the following simple linear regression model:
    Annual Income = -45,353 + 7,120*(Years of Schooling). This tells the economist
    that for each additional year of schooling, annual income increases by $7,120
    on average.
    • Biologists often collect bivariate data to understand how two variables are
    related among plants or animals. For example, a biologist may collect data

    on total rainfall and total number of plants in different regions:

    s

    The biologist may then decide to calculate the correlation between the two
    variables and find it to be 0.926. This indicates that there is a strong positive
    correlation between the two variables. That is, higher rainfall is closely
    associated with an increased number of plants in a region.
    • Researchers often collect bivariate data to understand what variables
    affect the performance of students. For example, a researcher may collect
    data on the number of hours studied per week and the corresponding GPA

    for students in a certain class:

    s

    She may then create a simple scatter plot to visualize the relationship between

    these two variables:

    d

    Clearly there is a positive association between the two variables: As the
    number of hours studied per week increases, the GPA of the student tends to

    increase as well.

    With the use of bivariate data we can quantify the relationship between
    variables related to promotions, advertising, sales, and other variables.
    Time series forecasting allows financial analysts to predict future revenue,

    expenses, new customers, sales, etc. for a variety of companies.

    Application activity 5.2.3

    A company is to replace its fleet of cars. Eight possible models are
    considered and the transport manager is asked to rank them, from 1 to 8,
    in order of preference. A saleswoman is asked to use each type of car for
    a week and grade them according to their suitability for the job (A-very

    suitable to E-unsuitable). The price is also recorded:

    x

    a) Calculate the Spearman’s coefficient of rank correlation between:

    i) Price and transport manager’s rankings,

    ii) Price and saleswoman’s grades.

    b) Based on the result of a, state, giving a reason, whether it would
    be necessary to use all the three different methods of assessing

    the cars.

    c) A new employee is asked to collect further data and to do some
    calculations. He produces the following results: The coefficient of

    correlation between

    i) Price and boot capacity is 1.2,

    ii) Maximum speed and fuel consumption in miles per gallons is -0.7,

    iii) Price and engine capacity is -0.9. For each of his results, say giving

    a reason, whether you think it is reasonable.

    d) Suggest two sets of circumstances where Spearman’s coefficient
    of rank correlation would be preferred to the Pearson’s coefficient

    of correlation as a measure of association.

    End of unit assessment 5

    1. Table below shows the marks awarded to six students in accounting

    competition:

    d

    Calculate a coefficient of rank correlation.

    2. At the end of a season, a league of eight hockey clubs produced the
    following table showing the position of each club in the league and

    the average attendance (in hundreds) at home matches.

    d

    Calculate Spearman’s coefficient of rank correlation between the position
    in the league and average attendance. Comment on your results.
    3. The following results were obtained from lineups in Accounting

    and Finance examinations:

    d

    Find both equations of regression lines. Also estimate the value of y for

    x=30.

    4. The following results were obtained from records of age (x) and

    systolic blood pressure Yes of a group of 10 men:

    d

    Find both equations of the regression lines. Also estimate the blood

    pressure of a man whose age is 45.

    REFERENCES

    1. Arem, C. (2006). Systems and Matrices. In C. A. DeMeulemeester, Systems
    and Matrices (pp. 567-630). Demana: Brooks/Cole Publishing 1993 &

    Addison-Wesley1994.

    2. Kirch, W. (Ed.). (2008). Level of MeasurementLevel of measurement BT
    - Encyclopedia of Public Health (pp. 851–852). Springer Netherlands.

    https://doi.org/10.1007/978-1-4020-5614-7_1971

    3. Crossman, Ashley. (2020). Understanding Levels and Scales of
    Measurement in Sociology. Retrieved from https://www.thoughtco.
    com/levels-of-measurement-3026703

    4. Markov. ( 2006). Matrix Algebra and Applications. In Matrix Algebra and

    Applications (pp. p173-208.).

    5. REB, R.E (2020). Mathematics for TTCs Year 2 Social Studies Education.

    Student’s book.

    6. REB, R.E (2020). Mathematics for TTCs Year 3 Social Studies Education.

    Student’s book

    7. REB, R.E (2020). Mathematics for TTCs Year 1 Science Mathematics

    Education. Students’ book

    8. REB, R.E (2020). Mathematics for TTCs Year 1 Social Studies Education.

    Student’s book

    9. Rossar, M. (1993, 2003). Basic Mathematics For Economists. London and

    New York: Taylor & Francis Group.

    ◄Unit 4 Univariate Statistics and ApplicationsTopic 6►