NTS NAT-IGS Statistics Syllabus

What is a relative frequency?

Relative frequency is the ratio of a class frequency to the total number of observations, often expressed as a fraction or percentage: f / n.

What is a cumulative frequency distribution?

A cumulative frequency distribution shows the running total of frequencies up to and including each class, indicating how many observations fall below a given boundary.

What is Sturges' rule for the number of classes?

Sturges' rule: number of classes k = 1 + 3.322 log₁₀(n), where n is the total number of observations.

What is the difference between classification and tabulation?

Classification is arranging data into groups or classes based on common characteristics; tabulation is presenting classified data in rows and columns of a table.

What are the main bases (types) of classification?

Data may be classified geographically (by place), chronologically (by time), qualitatively (by attribute), and quantitatively (by magnitude).

What is the difference between a simple table and a complex table?

A simple (one-way) table presents data by a single characteristic, while a complex table presents data classified by two or more characteristics simultaneously.

What are the essential parts of a statistical table?

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

What is the difference between a histogram and a bar diagram?

A histogram represents continuous (grouped) data using adjacent bars with no gaps, where area shows frequency; a bar diagram represents discrete/categorical data using separated bars whose height shows value.

What is a frequency polygon?

A frequency polygon is a line graph formed by joining the midpoints (class marks) of the tops of histogram bars, used to display a frequency distribution.

What is an ogive?

An ogive is a cumulative frequency curve obtained by plotting cumulative frequencies against class boundaries and joining the points with a smooth curve.

What is a pie chart (pie diagram)?

A pie chart is a circular diagram divided into sectors, where each sector's angle (and area) is proportional to the value of the category it represents.

How is the angle for a component in a pie chart calculated?

Angle = (Component value / Total value) × 360°.

What is the difference between a histogram and a frequency polygon?

A histogram uses bars (area) to show frequency, while a frequency polygon uses straight lines joining class midpoints; the polygon is better for comparing distributions.

Define the arithmetic mean and give its formula for ungrouped data.

The arithmetic mean is the sum of all values divided by the number of values: x̄ = Σx / n.

What is the formula for the arithmetic mean of grouped data?

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

Define the median.

The median is the middle value of an ordered data set that divides it into two equal halves; 50% of values lie below and 50% above it.

How do you find the median position for n ordered values?

The median is the value at position (n + 1)/2 when data is arranged in order; for even n it is the average of the two middle values.

Define the mode.

The mode is the value or category that occurs most frequently in a data set.

What is the empirical relationship between mean, median, and mode?

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

Which measure of central tendency is best for open-ended classes and why?

The median, because it depends only on the middle position and is not affected by extreme or undefined (open-ended) class limits.

Which average is most affected by extreme values?

The arithmetic mean is most affected by extreme values (outliers).

What is the formula for the geometric mean of n values?

Geometric Mean = ⁿ√(x₁ · x₂ · ... · xₙ), the nth root of the product of all n values; equivalently GM = antilog[(Σ log x)/n].

When is the geometric mean most appropriate?

The geometric mean is best for averaging ratios, rates of change, percentages, and growth rates (e.g., population or price index growth).

What is the formula for the harmonic mean of n values?

Harmonic Mean = n / Σ(1/x), the reciprocal of the arithmetic mean of the reciprocals of the values.

When is the harmonic mean most appropriate?

The harmonic mean is best for averaging rates and ratios such as speed (distance per time) when the units in the denominator are constant.

What is the relationship in size among the arithmetic, geometric, and harmonic means?

For positive values that are not all equal: AM ≥ GM ≥ HM (the arithmetic mean is largest and the harmonic mean is smallest).

What are quartiles?

Quartiles are three values (Q₁, Q₂, Q₃) that divide an ordered data set into four equal parts; Q₂ is the median.

What are deciles and percentiles?

Deciles are nine values (D₁–D₉) dividing ordered data into ten equal parts; percentiles are ninety-nine values (P₁–P₉₉) dividing it into one hundred equal parts.

What is the relationship between the median, Q₂, D₅, and P₅₀?

They are all equal — the median = second quartile (Q₂) = fifth decile (D₅) = fiftieth percentile (P₅₀).

What is the range and how is it calculated?

The range is the simplest measure of dispersion: Range = Largest value − Smallest value (Xmax − Xmin).

What is the quartile deviation (semi-interquartile range)?

Quartile Deviation = (Q₃ − Q₁) / 2; it measures the spread of the middle 50% of the data and ignores extreme values.

What is the interquartile range?

The interquartile range is the difference between the third and first quartiles: IQR = Q₃ − Q₁, covering the middle 50% of observations.

What is the mean deviation?

Mean deviation is the arithmetic mean of the absolute deviations of all values from a central value (usually the mean or median): MD = Σ|x − x̄| / n.

From which central value is the mean deviation the minimum?

The mean deviation is least (minimum) when deviations are measured from the median.

What is variance?

Variance is the average of the squared deviations of values from their mean: σ² = Σ(x − x̄)² / n.

What is the standard deviation and its formula?

Standard deviation is the positive square root of the variance: σ = √[Σ(x − x̄)² / n]. It measures the average spread of data about the mean.

Why is standard deviation often preferred over mean deviation?

Standard deviation uses squared deviations (avoiding the absolute-value problem), is algebraically tractable, and is used in further statistical analysis, making it the most reliable measure of dispersion.

What is the coefficient of variation and what does it measure?

Coefficient of Variation (CV) = (Standard Deviation / Mean) × 100. It is a relative, unit-free measure of dispersion used to compare variability between data sets; a lower CV indicates greater consistency.

What is the difference between population variance and sample variance formulas?

Population variance divides by N: σ² = Σ(x − μ)²/N, while sample variance divides by (n − 1): s² = Σ(x − x̄)²/(n − 1) to give an unbiased estimate.