Statistics

Statistics is the branch of mathematics that deals with the collection, presentation and interpretation of numerical facts, called data. Data can be of two kinds. Primary data is collected by the investigator personally for a specific purpose, such as recording the heights of students in your own class. Secondary data is data that already exists, collected by someone else for another purpose, such as figures published in a newspaper or a government report.

Data collected in its original form, before any arrangement, is called raw data (or ungrouped data). Raw data is hard to read at a glance, so it is organised using tally marks into a frequency distribution table, which shows how many times (the frequency) each value occurs.

Worked Example. The number of goals scored by a school football team in 12 matches were: 1, 2, 0, 1, 3, 2, 1, 0, 2, 1, 1, 2. Arrange this as an ungrouped frequency distribution table.

• Goals = 0: frequency 2
• Goals = 1: frequency 5
• Goals = 2: frequency 4
• Goals = 3: frequency 1

Check: 2 + 5 + 4 + 1 = 12, which matches the total number of matches, so the table is correct.

When the range of data (the difference between the maximum and minimum value) is large, individual values are grouped into class intervals of equal width, such as 0-10, 10-20, 20-30, and so on. Each class interval has a lower limit and an upper limit. In the commonly used exclusive form, the upper limit of a class is not included in that class — it is counted in the next class. The mid-value of a class is called its class mark, and it represents that class on a graph.

Worked Example. For the class interval 20-30, the class size is 30 − 20 = 10, and the class mark is (20 + 30) ÷ 2 = 25.

A bar graph is a graphical way of showing data using rectangular bars of equal width, with uniform gaps between consecutive bars. The height of each bar is proportional to the frequency or value it represents. Bar graphs are used for categorical data (such as blood groups) or discrete numerical data (such as the number of goals scored).

Worked Example. Using the goals data from before (0 goals: 2 matches, 1 goal: 5 matches, 2 goals: 4 matches, 3 goals: 1 match), a bar graph would have four bars of equal width along the horizontal axis labelled 0, 1, 2, 3, with heights 2, 5, 4 and 1 on the vertical (frequency) axis, each bar separated from the next by an equal gap.

A histogram is used only for continuous grouped data arranged in class intervals of equal width. Unlike a bar graph, the rectangles of a histogram touch each other, with no gap, because the class intervals are continuous. The height of each rectangle is simply the frequency of that class, and the width of each rectangle equals the class size, which is the same for every class.

Worked Example. For the grouped frequency table with equal class size 10 — 0-10: 3, 10-20: 5, 20-30: 6, 30-40: 7, 40-50: 4 — a histogram is drawn with five touching rectangles over these classes, with heights equal to 3, 5, 6, 7 and 4 respectively, since all class widths are equal.

A frequency polygon is another way of representing grouped frequency data using a line graph. It can be drawn in two ways: (i) by first drawing a histogram, then joining the midpoints of the tops of consecutive rectangles with straight lines, or (ii) directly, by plotting the class mark of each class against its frequency and joining the plotted points with straight line segments.

To complete the polygon so that it forms a closed figure with the horizontal axis, one extra class of zero frequency is imagined immediately before the first class and one extra class of zero frequency immediately after the last class. The polygon is then joined to the class marks of these two imaginary classes on the horizontal axis.

Worked Example. For the grouped frequency table 0-10: 3, 10-20: 5, 20-30: 6, 30-40: 7, 40-50: 4, the class marks are 5, 15, 25, 35 and 45. To close the polygon, an imaginary class (−10)-0 (class mark −5, frequency 0) is added before, and an imaginary class 50-60 (class mark 55, frequency 0) is added after. The frequency polygon then joins the points (−5, 0), (5, 3), (15, 5), (25, 6), (35, 7), (45, 4) and (55, 0) in order with straight lines.

Frequency polygons are especially useful for comparing two or more sets of grouped data on the same axes, since several polygons can be drawn on one graph while histograms cannot easily overlap.

The mean (also called the average), denoted x̄ (read as "x-bar"), is the most common measure of central tendency. For raw (ungrouped) data with n observations x₁, x₂, ..., x_n, the mean is found by adding all the observations and dividing by the number of observations.

Worked Example 1. A batsman scored 45, 60, 32, 78 and 50 runs in 5 innings. Find his mean score.

Sum of runs = 45 + 60 + 32 + 78 + 50 = 265.

Mean = 265 ÷ 5 = 53 runs.

When the same value occurs several times, the data can be given as an ungrouped frequency distribution — a table of distinct values x_i along with how many times each occurs, f_i. In this case, the mean is found using x̄ = Σ(f_i × x_i) ÷ Σf_i, where Σf_ix_i means "the sum of each value multiplied by its frequency", and Σf_i is the total number of observations.

Worked Example 2. The number of children in 10 families of a locality is given below.

• Children = 0: frequency 1
• Children = 1: frequency 2
• Children = 2: frequency 4
• Children = 3: frequency 2
• Children = 4: frequency 1

Check: Σf_i = 1 + 2 + 4 + 2 + 1 = 10 families, which matches the given total.

Σ(f_i × x_i) = (0×1) + (1×2) + (2×4) + (3×2) + (4×1) = 0 + 2 + 8 + 6 + 4 = 20.

Mean = 20 ÷ 10 = 2 children per family.

The median is the value of the middle-most observation when the raw data is arranged in ascending (or descending) order. To find it: arrange the n observations in order; if n is odd, the median is the value of the ((n + 1) ÷ 2)th observation; if n is even, the median is the average of the (n ÷ 2)th observation and the (n ÷ 2 + 1)th observation.

Worked Example 1 (odd n). The weights (in kg) of 7 students are 42, 38, 45, 33, 40, 36, 50. Arranged in ascending order: 33, 36, 38, 40, 42, 45, 50. Here n = 7 (odd), so the median is the ((7+1)÷2) = 4th observation, which is 40 kg.

Worked Example 2 (even n). The marks of 6 students are 55, 62, 48, 70, 58, 65. Arranged in ascending order: 48, 55, 58, 62, 65, 70. Here n = 6 (even), so the median is the average of the 3rd and 4th observations: (58 + 62) ÷ 2 = 60.

The mode of raw data is simply the observation that occurs the maximum number of times, that is, the value with the highest frequency. Data can have one mode (unimodal), more than one mode, or no repeated value at all.

Worked Example 3. The shoe sizes of 10 students are 6, 7, 8, 7, 6, 7, 9, 7, 8, 6. Counting how often each size occurs: size 6 occurs 3 times, size 7 occurs 4 times, size 8 occurs 2 times, and size 9 occurs 1 time. Since size 7 occurs the most often, the mode is 7.

The median and the mode are both measures of central tendency, like the mean, but unlike the mean, they are not affected much by one or two extremely high or low values in the data.