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Statistics

PRESENTING DATA
  • Raw data refers to data that has not undergone any form of processing; i.e., all the information is just congested and hard to make sense of.
  • Presenting data in some forms helps to ‘free up’ the data, i.e. make it more understandable.
  • Data can be presented using the following methods:
    • BAR CHART
      • A bar chart (or bar graph) is a chart (or graph) that represents categorical data with rectangular bars.
      • The lengths/heights of these bars are proportional to the values they represent.
      • The bar chart is usually drawn along the x-, and y-axis with the x-axis representing the categories of the data, the y-axis representing values for those categories.
      • Bar charts can be drawn horizontally or vertically. Vertically drawn bar charts are the most common.
      • A vertical bar chart is sometimes called a column chart.
      • Bar charts are used for discrete variables only.
      • When constructing a bar chart, it is important to choose a suitable scale to represent the frequency.
    • PIE CHART
      • A pie chart is a type of graph that displays data in a circular graph.
      • In a pie chart, the length of the arc of each slice (and consequently its central angle and area), is proportional to the quantity it represents.
      • In other words, each ‘slice’ of the pie is relative to the size of that category in the group as a whole.
      • The entire ‘pie’ represents 100 percent of a whole, while the ‘slices’ of the pie represent portions of the whole.
      • A pie chart gives you a snapshot of how a group is broken down into smaller pieces.
AVERAGES
  • The average of a set of numbers is a number that is representative of a whole set of numbers.
  • There are three common averages: the mean, median, and mode.
  • The mean is the sum of all the values divided by the number of values.
  • The median is the middle value when all the numbers are arranged in order of size. If there are two middle values, the mean is the mean of the two numbers.
  • The mode is the value of the highest frequency. Some distributions (sets of values) may have more than one mode. A bimodal distribution has two modes.
PROBABILITY
  • If there are n possible outcomes and r required outcomes, the probability, p, of obtaining a required outcome is given as a fraction:
    • Probability, P = Number of required outcomes/Number of possible outcomes = r/n
    • where:
      • P lies between 0 and 1, and
      • P may be expressed as a common fraction, a decimal fraction, or percentage.
  • If an outcome is certain to happen, its probability is 1.
  • If an outcome is not certain to happen, its probability is 0.
  • If the probability of an event happening is p, then the probability of the event not happening is 1 – p.
Examples
  1. In a Government College in the west of Nigeria, 400 students were asked what their mother language was. The results were as follows:
    Hausa55
    Igbo70
    Yoruba178
    Others97

    Represent this data with a bar chart.

    • For this question, we would pick a scale that would effectively house the frequencies given.
  2. The bar chart below shows the distribution of secondary school classes 1 – 6 in a school.

    1. How many students are in the school?
      • To get the total number of students in the school, we would add the total number of students in each class.
      • For class 1, the total number of students = 80
      • For class 2, the total number of students = 60
      • For class 3, the total number of students = 60
      • For class 4, the total number of students = 40
      • For class 5, the total number of students = 40
      • For class 6, the total number of students = 20
      • ∴ The total number  of students in the school = 80 + 60 + 60 + 40 + 40 + 20 = 300 students
    2. If each of the students gets three exercise books for the term, how many exercise books are needed?
      • Exercise books needed = Total number of students × Number of books needed by each student
      • ∴ Exercise books needed = 300 × 3
      • ∴ Exercise books needed = 900 books
  3. In a Government College in the west of Nigeria, 400 students were asked what their mother language was. The results were as follows:
    Hausa55
    Igbo70
    Yoruba178
    Others97

    Represent this data with a pie chart.

    • Firstly, we calculate the total number of students, which is 400.
    • Secondly, we calculate the ratio of number of students for each mother language to the total number of students, i.e.
      • For Hausa: 55/400 = 11/80
      • For Igbo: 70/400 = 7/40
      • For Yoruba: 178/400 = 89/200
      • For Others: 97/400 = 97/400
    • Thirdly, we multiply each of the ratios by the total angle of a pie (which is a circle), which is 360°
      • For Hausa: 11/80 × 360 = 49.5°
      • For Igbo: 7/40 × 360 = 63°
      • For Yoruba: 178/400 × 360 = 160.2°
      • For Others: 97/400 × 360 = 87.3°
      • NOTE: To check if you are correct, add the new angles together, they must be equal to 360°.
    • Fourthly, we plot the pie chart using the angles for each mother language.
  4. 72 fish were caught in a river in one day. The chart below shows the different kinds of fish caught.

    1. What percentage of fish were dogfish?
      • To calculate the percentage of fish that were dogfish, we would first divide the angle of the dogfish by the total angle of the pie (360°), and then multiply the result by 100%.
      • ∴ Percentage of fish that were dogfish = 90/360 × 100%
      • ∴ Percentage of fish that were dogfish = 25%
    2. What fraction of the fish were bream?
      • To calculate this fraction, we would divide the angle of the bream fish by the total angle of the pie.
      • ∴ The fraction of fish that were bream = 135/360 = 3/8
    3. How many kingfish were caught?
      • To calculate the amount of fish that were kingfish, we would find the fraction of the fish that were bream, and then multiply by the total number of fish that were caught.
      • ∴ Number of kingfish = 120/360 × 72 = 24 Kingfish.
    4. How many pike were caught?
      • Angle of pike fish = 360 – (120 + 90 + 135) = 15°
      • ∴ Number of pike fish = 15/360 × 72 = 3 Pike fish
  5. In five tests, a student’s marks were 13, 17, 18, 8, and 10. What is the average mark?
    • Average (Mean) mark = (13 + 17 + 18 + 8 + 10) ÷ 5
    •                                        = 66/5
    • ∴ The average mark = 13.2
  6. A hockey team has played eight games and has a mean score of 3.5 goals per game. How many goals has the team scored?
    • Mean score = Total number of goals ÷ Number of games
    • ∴ 3.5 = Total number of goals/8
    • Multiply both sides by 8: 3.5 × 8 = Total number of goals.
    • ∴ The total number of goals scored = 28 goals
  7. Find the median of 8.3, 11.3, 9.4, 13.8, 12.9, 10.5.
    • Arrange the set of numbers in order of size: 8.3, 9.4, 10.5, 11.3, 12.9, 13.8
    • There are six numbers, the median is the mean of the third and fourth numbers.
    • Median = (10.5 + 11.3) ÷ 2
    •            = 21.8/2
    • ∴ The median = 10.9
  8. In a class test, the marks scored by 40 students are shown in the table below. Find the mode of the marks.

    • The greatest frequency is 8.
    • Two marks have this frequency; 6 and 9.
    • Thus there are two modes; 6 and 9.
    • This distribution is said to be bimodal
  9. It is known that out of every 1000 new cars, 50 develop a mechanical fault in the first three months. What is the probability of buying a car that will develop a mechanical fault within three months?
    • Number of cars developing faults = 50
    • Number of cars altogether = 1000
    • Probability of buying a faulty car = 50/1000
    • ∴ P = 1/20
  10. A tray of eggs contains 18 large-sized and 12 small-sized eggs. An egg is selected at random. Find the probability of selecting:
    1. a small-sized egg.
      • The total number of eggs in the tray = 30
      • Number of small-sized eggs = 12
      • ∴ The probability of selecting a small-sized egg = 12/30
      • ∴ The probability of selecting a small-sized egg = 2/5
    2. either a small-sized or a large-sized egg.
      • It is certain that either a small-sized egg or a large-sized egg will be selected.
      • ∴ P = 30/30 = 1
    3. Neither a small-sized nor a large-sized egg.
      • The eggs are either large or small. It is impossible to select any other size.
      • ∴ P = 0/30 = 0
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