Statistics

Class mark

Class mark = (upper class limit + lower class limit) ÷ 2, the mid-point of a class. For grouped data it stands in for every observation in the class, since the class frequency is assumed to be centred there.

Direct method for the mean

x̄ = Σfᵢxᵢ ÷ Σfᵢ, where xᵢ is the class mark and fᵢ the frequency of each class.

Assumed mean method

x̄ = a + Σfᵢdᵢ ÷ Σfᵢ, where a is a class mark chosen as the assumed mean and dᵢ = xᵢ − a. Choosing a near the centre of the class marks keeps the numbers small.

Step-deviation method

x̄ = a + h × (Σfᵢuᵢ ÷ Σfᵢ), where uᵢ = (xᵢ − a)/h, a is the assumed mean and h the class size. It is handiest when all the deviations share the common factor h.

The same marks give a mean of 59.3 from the raw data but 62 once grouped. Why do they differ, and which is exact?

Grouping replaces every mark in a class by the class mark, assuming the observations are centred on the mid-point. That assumption is only approximately true, so 59.3 from the raw data is exact and 62 is an approximation.

How do you choose between the direct, assumed mean and step-deviation methods for a grouped mean?

All three give the same mean. Use the direct method when xᵢ and fᵢ are small. For large numbers use the assumed mean or step-deviation method; with unequal class sizes, step deviation still works if h is a common divisor of all the dᵢ.

For marks in classes of width 15 with class marks 17.5 to 92.5, how is the mean found by step deviation?

Take a = 47.5 and h = 15, and find uᵢ = (xᵢ − 47.5)/15 for each class, giving −2 to 3. Multiply by the frequencies and add to get Σfᵢuᵢ = 29 with Σfᵢ = 30. Then x̄ = 47.5 + 15 × 29/30 = 62.

Modal class

The class of a grouped frequency distribution that has the highest frequency. The mode of grouped data is a value inside this class, not simply its mid-point.

Mode of grouped data

Mode = l + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h, where l is the modal class's lower limit, h the class size, f₁ its frequency, and f₀ and f₂ the frequencies of the classes just before and after it.

For family sizes in classes 1-3, 3-5, 5-7, ... with frequencies 7, 8, 2, 2, 1, how do you find the mode?

The highest frequency 8 makes 3-5 the modal class, so l = 3, h = 2, f₁ = 8, f₀ = 7 and f₂ = 2. Then mode = 3 + (8 − 7)/(16 − 7 − 2) × 2 = 3 + 2/7 ≈ 3.286.

For the same marks data the mode is 52 and the mean is 62. What does each value tell you?

The mode says the largest number of students scored around 52, while the mean says that on average a student scored 62. Which one matters depends on whether you want the typical score of most students or the overall average.

Cumulative frequency

The running total of frequencies up to a class. In the chapter's less-than table, the class 10-20 has cumulative frequency 5 + 3 = 8, the frequencies of 0-10 and 10-20 added together.

Less than type and more than type cumulative frequency distributions

A less than type distribution counts observations below each upper class limit, such as 'less than 20'. A more than type distribution counts observations at or above each lower class limit, such as 'more than or equal to 10'.

Median class

Among the classes whose cumulative frequency exceeds n/2, the one whose cumulative frequency is closest to n/2, where n is the total number of observations. The median of grouped data lies inside this class.

Median of grouped data

Median = l + [(n/2 − cf) ÷ f] × h, where l is the median class's lower limit, n the number of observations, cf the cumulative frequency of the class before it, f its frequency and h the class size.

Heights of 51 girls are given as 'less than 140', 'less than 145', and so on. How do you find the median height?

Subtract consecutive cumulative counts to rebuild the class frequencies. Since n/2 = 25.5, the median class is 145-150, with l = 145, cf = 11, f = 18 and h = 5. So median = 145 + (25.5 − 11)/18 × 5 ≈ 149.03 cm.

A distribution with total frequency 100 has two missing frequencies x and y and median 525. How do you find them?

Add the known frequencies to get x + y = 24. The median 525 lies in 500-600, where cf = 36 + x and f = 20. Substituting into the median formula gives 25 = (14 − x) × 5, so x = 9 and y = 15.

Empirical relationship between mean, median and mode

3 Median = Mode + 2 Mean. It lets you estimate any one of the three measures when the other two are known.

When is the mean, the median or the mode the best measure of central tendency?

The mean uses every observation and suits comparisons, but extreme values distort it. The median suits finding a typical value when extremes are present, such as wages. The mode suits finding the most popular item, such as the best-selling product.

Before using the median or mode formula on classes like 118-126, 127-135, why must you adjust them, and how?

The formulas assume continuous classes with no gaps between them. Close each gap by moving the limits half-way, so 118-126 and 127-135 become 117.5-126.5 and 126.5-135.5, then apply the formula.

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