Basic statistics: Class 10 Mathematics

Key points

Statistics is the science of collecting, organising, presenting and interpreting data. Raw data are the original observations. Data can be discrete, such as the number of children in a family, or continuous, such as height or weight. A set of data arranged in order is called an array.

A frequency distribution groups data into classes and shows how many observations fall in each class. The number of observations in a class is its frequency f. A tally mark is used to count them. The class width h is the difference between the upper and lower class boundaries. The class mark (mid point) is the average of the lower and upper limits of a class.

A measure of central tendency gives a single value that represents the whole data. The most used measures are the arithmetic mean, the median and the mode. A good average is clearly defined, easy to calculate and uses all observations.

Arithmetic mean:

• Ungrouped data: mean = Σx / n, the sum of observations divided by their number.
• Grouped data, direct method: mean = Σfx / Σf, where x is the class mark.
• Short cut method: take an assumed mean A and deviations D = x − A. Then mean = A + ΣfD / Σf.
• Coding method: with u = (x − A)/h, mean = A + (Σfu / Σf) × h.

The weighted mean is used when observations have different importance. If x₁, x₂, ... have weights w₁, w₂, ..., the weighted mean is Σwx / Σw. It is used in finding an overall grade or an average price.

The median is the middle value when data are arranged in order. For n values, if n is odd the median is the ((n + 1)/2)th value. If n is even, it is the average of the (n/2)th and (n/2 + 1)th values. The median is not affected by very large or very small values, so it suits skewed data such as incomes.

The mode is the value that occurs most often. Data can have no mode, one mode, or more than one mode. For grouped data the class with the highest frequency is the modal class. The mode is the only average that can be used for qualitative data, such as the most popular shirt colour.

The three averages are related for moderately skewed data by the approximate rule: mode ≈ 3 median − 2 mean. In a symmetrical distribution, the mean, median and mode are equal.

A measure of dispersion tells how far the data are spread out. Two sets of data can have the same mean but different spread. The range is the simplest measure: range = largest value − smallest value. The coefficient of range is (L − S)/(L + S). The range uses only two values and is affected by extreme values.

Variance and standard deviation:

• Variance σ² = Σ(x − mean)² / n for ungrouped data.
• Standard deviation σ = √variance. It has the same unit as the data.
• Computing form: σ² = Σx²/n − (Σx/n)².
• Grouped data: σ² = Σfx²/Σf − (Σfx/Σf)².
• Coefficient of variation = (σ / mean) × 100%, used to compare the spread of two data sets.

Study tip: Show your working in a table with columns for x, f, fx and, for dispersion, fx² or the squared deviations. A table makes errors easy to see and earns method marks. Always check that the mean lies between the smallest and largest value, and give the unit with the standard deviation.

Definitions and flashcards

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Formula sheet

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Short questions with model answers

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Numericals with worked steps

Common mistakes

• Using the class limits instead of the class marks when finding the mean of grouped data.
• Finding the median without arranging the data in order first.
• Forgetting to take the square root, and giving the variance as the standard deviation.
• Dividing by Σw or n wrongly in the weighted mean, or dividing by the number of classes instead of Σf.
• Saying there is no mode when two values have the same highest frequency. Such data are bimodal.

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