ICS (Intermediate in Computer Science) Statistics (Part I) - Elective Syllabus

Define classification of data.

Classification is the process of arranging data into groups or classes according to their common characteristics or resemblances.

Name the four bases (types) of classification of data.

Qualitative, quantitative, geographical (spatial), and chronological (temporal) classification.

What is chronological classification?

Classification of data according to time (e.g., production figures arranged year by year).

Define tabulation of data.

Tabulation is the systematic arrangement of classified data into rows and columns of a table for clear presentation and comparison.

Name the essential parts of a statistical table.

Table number, title, head note, captions (column headings), stubs (row headings), body, and footnote/source note.

What is the difference between a stub and a caption in a table?

Stubs are the headings/titles of the rows (left side); captions are the headings/titles of the columns (top).

Define a frequency distribution.

A frequency distribution is a tabular arrangement of data showing the classes (or values) and the number of observations (frequencies) falling in each class.

Define class limits.

Class limits are the smallest (lower) and largest (upper) values that a class can include, e.g., in 10–19 the lower limit is 10 and upper limit is 19.

Define class boundaries.

Class boundaries are the true (real) limits of a class obtained by removing the gap; e.g., class 10–19 has boundaries 9.5–19.5.

How is the class mark (midpoint) of a class calculated?

Class mark = (lower class limit + upper class limit) / 2, i.e., the average of the two limits (or two boundaries).

How is the class interval (size of class) found?

Class interval = upper class boundary − lower class boundary (the difference between the boundaries of a class).

Give the formula for the approximate number of classes (Sturges' rule).

k = 1 + 3.322 log₁₀ n, where n is the number of observations.

How is the class width (h) estimated from range?

h = Range / Number of classes = (Largest value − Smallest value) / k.

What is a cumulative frequency distribution?

A distribution showing the running total of frequencies, i.e., the number of observations less than (or more than) the upper boundary of each class.

What is relative frequency?

Relative frequency of a class = class frequency / total frequency (f/Σf); it shows the proportion of observations in that class.

What is a histogram?

A graph of a frequency distribution made of adjacent rectangles, with class boundaries on the x-axis and frequencies as heights, for continuous (grouped) data.

What is a frequency polygon?

A line graph of a frequency distribution obtained by plotting class marks against frequencies and joining the points with straight lines.

What is an ogive?

An ogive is a cumulative frequency curve, obtained by plotting cumulative frequencies against upper (or lower) class boundaries and joining the points.

Differentiate between a histogram and a bar chart.

A histogram has adjacent bars (no gaps) representing continuous/grouped data with area meaning frequency; a bar chart has separated bars representing discrete or qualitative data with length showing magnitude.

What is a pie chart (sector diagram)?

A circular chart divided into sectors where each sector's angle (= component/total × 360°) is proportional to the value of the category it represents.

Define the arithmetic mean.

The arithmetic mean is the sum of all values divided by the number of values; it is the most common average.

State the formula for the arithmetic mean of ungrouped data.

x̄ = Σx / n, where Σx is the sum of all observations and n is their number.

State the formula for the arithmetic mean of grouped/frequency data.

x̄ = Σfx / Σf, where x is the class mark, f is the frequency, and Σf = n.

State an important mathematical property of the arithmetic mean regarding deviations.

The algebraic sum of the deviations of the observations from their mean is zero: Σ(x − x̄) = 0.

Define the median.

The median is the middle value of a data set arranged in order of magnitude; it divides the data into two equal halves.

How do you find the median position for ungrouped data?

Arrange data in order; the median is the value at position (n + 1)/2 (for odd n) or the average of the two middle values (for even n).

State the formula for the median of grouped data.

Median = l + (h/f)((n/2) − C), where l = lower boundary of median class, h = class width, f = frequency of median class, C = cumulative frequency before it, n = Σf.

Define the mode.

The mode is the value (or class) that occurs most frequently in a data set; the value with the highest frequency.

State the empirical relationship between mean, median, and mode.

Mode = 3 Median − 2 Mean (for a moderately skewed distribution).

Define the geometric mean.

The geometric mean of n positive values is the nth root of their product: G.M. = (x₁·x₂·…·xₙ)^(1/n); useful for averaging ratios and rates of change.

Define the harmonic mean.

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals: H.M. = n / Σ(1/x); useful for averaging rates/speeds.

State the size relationship among the arithmetic, geometric, and harmonic means.

For positive values, A.M. ≥ G.M. ≥ H.M., with equality only when all values are equal.

What are quantiles?

Quantiles are values that divide an ordered data set into a number of equal parts (e.g., quartiles into 4, deciles into 10, percentiles into 100 parts).

What are quartiles, and what do Q1, Q2, Q3 represent?

Quartiles divide ordered data into four equal parts: Q1 (lower) marks 25%, Q2 (median) 50%, and Q3 (upper) 75% of observations.

How many deciles and percentiles are there?

There are 9 deciles (D1 to D9) dividing data into 10 equal parts, and 99 percentiles (P1 to P99) dividing data into 100 equal parts.

Give the formula for Q1 (first quartile) of grouped data.

Q1 = l + (h/f)((n/4) − C), where l = lower boundary of Q1 class, h = class width, f = frequency of the class, C = cumulative frequency before it.

What relationships hold between quartiles, deciles, percentiles, and the median?

Q2 = D5 = P50 = Median; also D1 = P10, Q1 = P25, Q3 = P75 (the same positional values).

Define the range as a measure of dispersion.

Range = Largest value (Xmax) − Smallest value (Xmin); it is the simplest measure of dispersion showing the spread of data.

What is the coefficient of range?

Coefficient of Range = (Xmax − Xmin) / (Xmax + Xmin); a relative (unit-free) measure of dispersion based on range.

Define the quartile deviation (semi-interquartile range).

Quartile Deviation = (Q3 − Q1) / 2, i.e., half the interquartile range; it measures the spread of the middle 50% of data.