Chapter 14 — Statistics

What is Statistics? Statistics is the science of collecting, organising, presenting, analysing and interpreting numerical data to make decisions or draw conclusions. It converts raw data into useful information.

Why study Statistics? To summarise large amounts of data compactly, to compare groups, to identify patterns or trends, and to make informed decisions in fields such as education, business, health, government and science.

Basic steps in a statistical study

• Collection of data (primary or secondary)
• Organisation and classification (tables, frequency distributions)
• Presentation (graphs and charts)
• Analysis (measures of central tendency and dispersion)
• Interpretation and conclusion

Types of data

• Primary data: collected first-hand (surveys, experiments)
• Secondary data: obtained from existing sources (books, reports)
• Qualitative (categorical) vs Quantitative (numerical)
• Discrete (countable values) vs Continuous (measured values)

Raw data and frequency distribution: When many observations exist, we group them into classes (class intervals) and record frequencies (fi). For grouped data use class boundaries and class mark (xi) = (lower limit + upper limit)/2. Cumulative frequency (cf) is the running total of frequencies up to a class.

Measures introduced in Class 10: Measures of central tendency — mean, median and mode — are used to describe the centre of a data set. For grouped data we use class marks and standard formulas (including the assumed mean method) to compute these measures.

Important points

• Choose appropriate class width (equal width classes are usual).
• Label axes and give a title when drawing graphs.
• Use cumulative frequency (ogive) to read median graphically.

What is Frequency Distribution?
A frequency distribution is a way to organize raw data into classes or categories showing how often (frequency) each value or class occurs. It simplifies large data sets to reveal patterns (like peaks, spread) and is the first step in making statistical graphs.

Types

• Ungrouped frequency distribution — used for discrete or small data sets; lists each distinct value with its frequency.
• Grouped frequency distribution — used for large or continuous data; data are grouped into class intervals (e.g., 10–14, 15–19) with frequency for each class.

Important terms

• Class limits — lower and upper values of a class (e.g., 10 and 14)
• Class boundaries — true boundary values used for continuous data (often lower limit − 0.5 and upper limit + 0.5 when data are whole numbers)
• Class width (size) — difference between upper and lower limits of a class
• Class mark (mid-point) — (lower limit + upper limit) / 2
• Frequency (f) — number of observations in that class
• Cumulative frequency (CF) — running total of frequencies up to a class
• Relative/percentage frequency — frequency divided by total observations (or ×100 for percent)

How to construct a grouped frequency distribution (steps)

1. Find the range = maximum − minimum.
2. Choose number of classes (k). Typical choice: 5–15 classes; Sturges' rule gives k ≈ 1 + 3.3 log10 N as a guideline.
3. Compute class width ≈ range / k and round to a convenient value.
4. Set non-overlapping class intervals of equal width, then tally and count frequencies.

When preparing tables — always label columns (class limits/intervals, class mark, frequency, cumulative frequency, relative frequency).

Uses
Frequency distributions are used to summarize exam marks, heights, incomes, ages, temperatures, etc., and are the base for histograms, frequency polygons and ogives.

Graphical Representation of Data is the process of converting numerical data (raw or tabulated) into visual forms so patterns, trends and comparisons become easy to see and interpret. In Class 10 Statistics the common graphs are: bar graph, histogram, frequency polygon, and ogive (cumulative frequency graph). These represent frequency distributions (ungrouped or grouped).

Key steps before graphing: collect raw data, decide if you need an ungrouped or grouped frequency distribution, create class intervals (for grouped data), compute class-marks (mid-points) and frequencies, and compute cumulative frequencies if you will draw an ogive.

Choice of graph and main features:

• Bar graph: for categorical or discrete numerical data. Bars are separated by gaps; height ∝ frequency.
• Histogram: for continuous grouped data. Rectangular bars touch each other; width = class width and area ∝ frequency.
• Frequency polygon: join mid-points (class-marks) of top of histogram bars by straight lines; good for comparing distributions.
• Ogive (less-than or more-than): plot cumulative frequency against class boundary (or upper/lower class limits) to find median, quartiles and percentiles graphically.

Interpretation: read frequencies, proportions, central tendency (mean/median/mode) and spread from graphs; use ogive to estimate median and percentiles; compare distributions using shape (symmetry, skewness) and peaks (modes).

The arithmetic mean (commonly called the average) of a data set is a measure of central tendency that represents the centre or typical value of the data. For a set of numerical observations, the mean is obtained by adding all observations and dividing by the number of observations. The mean gives a single value that summarises the data but is sensitive to extreme values (outliers).

Types and methods:

• Ungrouped data (individual observations): Mean = (Sum of observations) / (Number of observations).
• Grouped data (frequency distribution): Use class mid-points. Mean = (Sum of frequency × mid-point) / (Total frequency).
• Assumed mean / Step-deviation methods: Useful to simplify calculations for grouped data when class mid-points are large. These methods reduce arithmetic by using an assumed mean A and, in the step-deviation method, dividing deviations by class width h.

Important properties to remember:

• Sum of deviations from the mean is zero: Σ(x - x̄) = 0.
• Mean is unique for a given data set.
• Mean is affected by extreme values (outliers).
• Linear transformation: If y = a x + b, then mean(y) = a·mean(x) + b.

When to use mean: for numerical data where you need a measure of central value and when the data have no large outliers. For skewed data or data with outliers, median may be preferred.

Definition: The median of a data set is the value that divides the ordered data into two equal parts — half the observations are less than or equal to it and half are greater than or equal to it. It is a measure of central tendency and is robust to outliers.

For ungrouped (individual) data:

• Arrange the data in ascending order.
• If the number of observations n is odd, the median is the value at position (n+1)/2.
• If n is even, the median is the average of the two middle values (positions n/2 and n/2 + 1).

Example (ungrouped): Data: 7, 3, 5, 9, 10. Ordered: 3, 5, 7, 9, 10. n = 5 (odd), median = value at (5+1)/2 = 3rd position = 7.

For grouped frequency distributions (continuous classes): we find the median class — the class where cumulative frequency first becomes ≥ N/2 (N = total frequency) — then use linear interpolation inside that class assuming uniform distribution of observations within the class.

Grouped data formula:

Median = l + ((N/2 − c.f.) / f) × h

where

• l = lower boundary of the median class (use true class boundary, e.g., 9.5 for class 10–19)
• N = total frequency
• c.f. = cumulative frequency of the class preceding the median class
• f = frequency of the median class
• h = width (class size) of the median class

Worked grouped example: Classes: 0–10: 5, 10–20: 8, 20–30: 12, 30–40: 5. N = 5+8+12+5 = 30. N/2 = 15. Cumulative freqs: 5, 13, 25, 30. Median class is 20–30 (cumulative becomes ≥ 15 here). l = 20, c.f. (before median class) = 13, f = 12, h = 10. Median = 20 + ((15 − 13)/12) × 10 = 20 + (2/12) × 10 ≈ 21.67.

Notes:

• Use true class boundaries when classes are continuous (e.g., 10–20 has boundaries 9.5 and 20.5 if data are continuous).
• Open-ended classes (like "50 and above") prevent exact median calculation unless additional information is given.
• Median is preferred over mean when data are skewed or contain outliers because it is less affected by extreme values.

Definition: The mode of a data set is the observation(s) that occur(s) most frequently. It is a measure of central tendency representing the most common value(s).

Types and existence: A data set may be unimodal (one mode), bimodal (two modes), multimodal (more than two), or have no mode if all values occur with the same frequency.

Mode for ungrouped (raw) data: For a list of individual observations, count frequencies and pick the value(s) with the highest frequency. Example: in {4, 7, 2, 4, 9, 4, 7} the mode is 4 (occurs 3 times).

Mode for discrete frequency distribution (tables): Identify the class or value with maximum frequency; that value (or class midpoint for grouped presentation of discrete values) is the mode.

Mode for continuous grouped data (class intervals): If data are given in class intervals with frequencies, the modal class is the class interval with the highest frequency. The mode can be estimated using the formula below (step-by-step):

1. Find the modal class (class with largest frequency).
2. Let l = lower boundary of the modal class (use class boundary, not the discrete limit), fm = frequency of modal class, f1 = frequency of the class before modal class, f2 = frequency of the class after modal class, and h = class width (upper boundary − lower boundary).
3. Apply the formula: Mode = l + ((fm - f1) / (2fm - f1 - f2)) × h.

Notes: For continuous grouped data you must use class boundaries (e.g., for class 10–20 boundaries are 9.5 and 20.5 if class limits are inclusive). The formula gives an estimate of the most likely location of the mode inside the modal class. If two or more classes share the maximum frequency, the distribution may be bimodal or multimodal and the formula is not directly applicable.

Interpretation and use: Mode is useful for categorical data (most common category), for identifying the most typical value in a dataset (e.g., most common shoe size, most sold product), and when the most frequent occurrence is more informative than average values. It is not affected by extreme values but may be unstable for small samples.

Overview: Mean, median and mode are measures of central tendency. For symmetric distributions they are equal. For skewed distributions they follow a typical order: in a positively skewed (right-skewed) distribution, Mode < Median < Mean; in a negatively skewed (left-skewed) distribution, Mean < Median < Mode.

Empirical (Pearson) relation: For many moderately skewed frequency distributions an approximate relation holds:

Mode ≈ 3 × Median − 2 × Mean

This relation comes from Pearson's coefficients of skewness and is an empirical (approximate) formula — it is useful for quick checks but is not an exact identity for every dataset.

Alternate forms:

• Mean − Mode ≈ 3 × (Mean − Median)
• Skewness (Pearson's second) ≈ 3(Mean − Median)/SD (where SD is standard deviation)

When to use and interpretation: Use the empirical relation when the distribution is unimodal and not highly irregular. If Mean > Median the data is right-skewed (tail to right); if Mean < Median the data is left-skewed (tail to left). The mode is the peak of the distribution and tends to be pulled opposite to the tail.

Grouped data formulas (to compute each quantity):

• Mean (using assumed mean a): Mean = a + (Σf d)/Σf, where d = x − a for class marks x and f = frequency.
• Median (grouped): Median = L + ((N/2 − C_f)/f_m) × h, where L = lower boundary of median class, N = total frequency, C_f = cumulative frequency before median class, f_m = frequency of median class, h = class width.
• Mode (grouped): Mode = L + ((f_m − f_1)/(2f_m − f_1 − f_2)) × h, where f_m = frequency of modal class, f_1 and f_2 are frequencies of the preceding and succeeding classes respectively, L = lower boundary of modal class, h = class width.

Important note: The Pearson relation is a rule of thumb. For multimodal distributions, or when the distribution is highly skewed or irregular, the relation may not hold well.

What it means: Practical applications and interpretation in statistics means using collected data (scores, incomes, measurements, counts) to calculate summary measures and to draw meaningful conclusions for decision making. It involves choosing appropriate measures (mean, median, mode), visualizing data (graphs), checking spread (range, quartiles, standard deviation) and being aware of limitations (outliers, biased samples, misleading graphs).

How to approach a real dataset:

• Step 1: Identify type of data — quantitative (continuous or discrete) or qualitative (categorical).
• Step 2: Choose summary measures — mean (typical numeric average), median (middle value, robust to outliers), mode (most frequent, useful for category and discrete data).
• Step 3: Measure spread — range, interquartile range (IQR), variance and standard deviation to understand variability.
• Step 4: Visualize — histogram/frequency polygon for distributions, ogive for medians/percentiles, box‑plot for spread and outliers, bar/pie charts for categorical proportions.
• Step 5: Interpret in context — compare groups, note skewness, consider effects of extreme values, and check if data collection was unbiased.

Key interpretation guidelines:

• Use median instead of mean when data are skewed or contain outliers (e.g., incomes).
• Use mode for categorical or most‑likely outcomes (e.g., most common shoe size).
• If two groups have similar means, compare spread (standard deviation or IQR) to decide which is more consistent.
• Ogives let you read medians, quartiles and percentiles directly — useful for ranking and cutoff decisions.
• Watch graph axes and scales — truncated axes or unequal class widths can mislead.

Common pitfalls: Small or biased samples give unreliable conclusions. Outliers can distort the mean. Grouped data require class‑wise methods (midpoints, class boundaries) for accurate estimates.