{"slug":"cbse-class-10-maths-statistics","title":"Statistics","description":"Mean, mode and median of grouped data and cumulative frequency, weighted towards the median-class and modal-class questions and the mean, mode and median calculations set in every readable 2026 paper.","board":"CBSE","grade":"Class 10","subject":"Maths","chapter":"Statistics","url":"https://www.flipnlearn.app/deck/cbse-class-10-maths-statistics","studyUrl":"https://www.flipnlearn.app/deck/cbse-class-10-maths-statistics/study","cards":[{"id":"class-mark","kind":"formula","front":"Class mark","back":"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."},{"id":"direct-method-for-the-mean","kind":"formula","front":"Direct method for the mean","back":"x̄ = Σfᵢxᵢ ÷ Σfᵢ, where xᵢ is the class mark and fᵢ the frequency of each class."},{"id":"assumed-mean-method","kind":"formula","front":"Assumed mean method","back":"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."},{"id":"step-deviation-method","kind":"formula","front":"Step-deviation method","back":"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."},{"id":"why-a-grouped-mean-is-approximate","kind":"concept-question","front":"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?","back":"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."},{"id":"which-method-for-the-mean","kind":"concept-question","front":"How do you choose between the direct, assumed mean and step-deviation methods for a grouped mean?","back":"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."},{"id":"how-to-find-the-mean-by-step-deviation","kind":"worked-step","front":"For marks in classes of width 15 with class marks 17.5 to 92.5, how is the mean found by step deviation?","back":"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."},{"id":"modal-class","kind":"term-definition","front":"Modal class","back":"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."},{"id":"mode","kind":"formula","front":"Mode of grouped data","back":"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."},{"id":"how-to-find-the-mode-of-grouped-data","kind":"worked-step","front":"For family sizes in classes 1-3, 3-5, 5-7, ... with frequencies 7, 8, 2, 2, 1, how do you find the mode?","back":"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."},{"id":"what-the-mean-and-mode-each-tell-you","kind":"concept-question","front":"For the same marks data the mode is 52 and the mean is 62. What does each value tell you?","back":"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."},{"id":"cumulative-frequency","kind":"term-definition","front":"Cumulative frequency","back":"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."},{"id":"less-than-and-more-than-type-distributions","kind":"term-definition","front":"Less than type and more than type cumulative frequency distributions","back":"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'."},{"id":"median-class","kind":"term-definition","front":"Median class","back":"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."},{"id":"median","kind":"formula","front":"Median of grouped data","back":"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."},{"id":"how-to-find-the-median-from-a-less-than-table","kind":"worked-step","front":"Heights of 51 girls are given as 'less than 140', 'less than 145', and so on. How do you find the median height?","back":"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."},{"id":"how-to-find-missing-frequencies-from-the-median","kind":"worked-step","front":"A distribution with total frequency 100 has two missing frequencies x and y and median 525. How do you find them?","back":"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."},{"id":"empirical-relationship","kind":"formula","front":"Empirical relationship between mean, median and mode","back":"3 Median = Mode + 2 Mean. It lets you estimate any one of the three measures when the other two are known."},{"id":"which-measure-of-central-tendency-to-use","kind":"concept-question","front":"When is the mean, the median or the mode the best measure of central tendency?","back":"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."},{"id":"why-classes-must-be-continuous","kind":"concept-question","front":"Before using the median or mode formula on classes like 118-126, 127-135, why must you adjust them, and how?","back":"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."}]}