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Formula Sheet
Statistics is the chapter that deals with the collection, analysis, interpretation, presentation, and organization of data.
Statistics – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the Statistics chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
Mean (Direct Method): x̄ = Σfᵢxᵢ / Σfᵢ
x̄ is the mean, fᵢ is the frequency of the ith observation, and xᵢ is the ith observation. This formula calculates the average of grouped data by summing the products of frequencies and their corresponding observations divided by the total frequency.
Mean (Assumed Mean Method): x̄ = a + (Σfᵢdᵢ / Σfᵢ)
a is the assumed mean, dᵢ = xᵢ - a. This method simplifies calculations by assuming a mean close to the data's center, adjusting the result based on deviations from this assumed mean.
Mean (Step Deviation Method): x̄ = a + (Σfᵢuᵢ / Σfᵢ) × h
uᵢ = (xᵢ - a)/h, where h is the class width. This method further simplifies the assumed mean method by scaling deviations, making calculations easier for large data sets.
Mode: Mode = l + ((f₁ - f₀) / (2f₁ - f₀ - f₂)) × h
l is the lower limit of the modal class, f₁ is the frequency of the modal class, f₀ and f₂ are frequencies of the preceding and succeeding classes, h is the class width. This formula finds the most frequently occurring value in grouped data.
Median: Median = l + ((n/2 - cf) / f) × h
l is the lower limit of the median class, n is the total number of observations, cf is the cumulative frequency before the median class, f is the frequency of the median class, h is the class width. This formula locates the middle value in grouped data.
Cumulative Frequency: CF = Σfᵢ up to the current class
CF is the cumulative frequency, fᵢ is the frequency of the ith class. It sums frequencies up to a certain point, useful for finding medians and quartiles.
Class Mark: (Lower Limit + Upper Limit) / 2
Represents the midpoint of a class interval, used as a representative value for calculations in grouped data.
Range: Maximum Value - Minimum Value
Measures the dispersion of data by calculating the difference between the highest and lowest values.
Empirical Relationship: 3 Median = Mode + 2 Mean
Links the three measures of central tendency, useful for approximating one measure when the other two are known.
Frequency Density: Frequency / Class Width
Adjusts frequency for class width, allowing comparison between classes of different sizes in histograms.
Equations
Mean of Ungrouped Data: x̄ = Σxᵢ / n
x̄ is the mean, xᵢ is the ith observation, n is the number of observations. Calculates the average of ungrouped data by summing all observations and dividing by their count.
Median of Ungrouped Data (Odd n): Median = ((n + 1)/2)th term
Finds the middle value in an ordered list of observations when the count is odd.
Median of Ungrouped Data (Even n): Median = (n/2th term + (n/2 + 1)th term) / 2
Averages the two middle values in an ordered list when the observation count is even.
Mode of Ungrouped Data: Most Frequently Occurring Observation
Identifies the value that appears most often in a data set, useful for categorical data analysis.
Cumulative Frequency Less Than Type: CF = Σfᵢ up to upper limit
Sums frequencies of all classes below a certain upper limit, used for constructing ogives.
Cumulative Frequency More Than Type: CF = Σfᵢ from lower limit
Sums frequencies of all classes above a certain lower limit, another method for ogive construction.
Step Deviation for Mean: uᵢ = (xᵢ - a) / h
Transforms observations into a simpler scale by subtracting an assumed mean and dividing by class width, facilitating easier calculation.
Class Interval Adjustment for Median: Exact Limits
Adjusts class limits for continuous data by subtracting 0.5 from lower limits and adding 0.5 to upper limits, ensuring accurate median calculation.
Frequency Polygon Midpoint: (Previous Class Mark + Next Class Mark) / 2
Calculates the x-coordinate for plotting frequency polygons, connecting midpoints of class intervals.
Histogram Area: Frequency Density × Class Width
Ensures the area of each bar represents the frequency, accommodating varying class widths in histograms.
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