Basic statistics: Class 9 Mathematics

Key points

Statistics is the science of collecting, organising, presenting and interpreting numerical data. Data are facts or figures gathered for a purpose, such as the marks of students or the daily rainfall of a city. Raw data (ungrouped data) are the values as first collected, without any arrangement.

Data are of two main kinds. Qualitative data describe quality, such as eye colour or favourite sport. Quantitative data are numbers. Quantitative data are discrete if they take only separate values, such as the number of children in a family, and continuous if they can take any value in a range, such as height or time.

Raw data are organised by arranging them in order, which is called an array, or by making a frequency distribution. A tally mark is made for each value (four vertical strokes crossed by a fifth makes a group of five), and the count for each value is its frequency f. The total of all frequencies is the number of observations n = Σf.

For a large amount of data, values are grouped into classes. A class interval is a range such as 10 to 19 or 20 to 29. The smallest and largest values of a class are its class limits. The class boundaries lie halfway between the upper limit of one class and the lower limit of the next: for classes 10 to 19 and 20 to 29 the boundary is 19.5, so the first class has boundaries 9.5 to 19.5.

Terms used with grouped data:

• Class width (class size) = upper boundary − lower boundary, for example 19.5 − 9.5 = 10.
• Class mark (mid-point) = (lower limit + upper limit) / 2, for example (10 + 19)/2 = 14.5.
• Range = largest value − smallest value of the data.
• Number of classes is usually between 5 and 15, and the classes should not overlap.

A histogram shows a grouped frequency distribution with bars. Class boundaries are marked on the horizontal axis and frequencies on the vertical axis. The bars touch each other, because the data are continuous. For equal class widths the height of a bar is its frequency. A histogram is different from a bar chart, where bars are separated and show separate categories.

A frequency polygon is drawn by plotting the class marks against the frequencies and joining the points with straight lines. It is usually closed at the ends by adding a class with zero frequency at each end, so the polygon touches the horizontal axis. It can also be drawn by joining the midpoints of the tops of the bars in a histogram.

A measure of central tendency is a single value that represents the whole data. The three common measures are the mean, the median and the mode. They show where the data are centred.

The arithmetic mean is the sum of all observations divided by their number. For raw data x̄ = Σx / n. For example, the mean of 4, 8, 6, 10, 12 is 40/5 = 8. For a frequency distribution, x̄ = Σfx / Σf, where x is the value or the class mark. The mean uses every value, but it is changed by very large or very small values.

The median is the middle value when data are arranged in order. For n values, if n is odd, the median is the value at position (n + 1)/2. If n is even, it is the average of the values at positions n/2 and n/2 + 1. For 3, 5, 7, 9, 12, 15, the middle two are 7 and 9, so the median is 8. The median is not much affected by extreme values.

The mode is the value that occurs most often. A data set may have no mode, one mode (unimodal), or two modes (bimodal). The mode of 2, 3, 3, 5, 5, 5, 7, 7 is 5. For qualitative data, such as the most common shoe size sold, only the mode makes sense.

The three measures give different information. Use the mean for symmetric data with no extreme values, the median when there are extreme values (such as incomes), and the mode for the most popular or most common item. When the mean changes by adding a new value, use the total: new total = old total + new value.

Study tip: Draw a neat frequency table with columns for class, class mark, frequency and fx, and check that the total of the frequencies equals the number of observations. For graphs, label both axes, write the scale and give the graph a title. For the median, always arrange the data in order first and show that step.

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Common mistakes

• Forgetting to arrange the data in order before finding the median.
• Using class limits instead of class boundaries when drawing a histogram, which leaves gaps between bars.
• Dividing Σfx by the number of classes instead of by Σf.
• Taking the highest frequency as the mode instead of the value that has that frequency.
• Using the class interval itself rather than the class mark as x when finding the grouped mean.

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