BSc CSIT (TU) Science Statistics I (BSc CSIT, STA164) Question Paper 2077 Nepal

Types of Data

By source:

• Primary data — collected first-hand by the investigator for the specific purpose (original data).
• Secondary data — already collected by someone else and reused (e.g. census reports, journals).

By nature/measurement:

• Qualitative (categorical) data — attributes that cannot be measured numerically (e.g. gender, religion).
• Quantitative (numerical) data — measurable, further split into discrete (countable, e.g. number of students) and continuous (any value in a range, e.g. height).
• Also classified by measurement scale: nominal, ordinal, interval, ratio.

Methods of Data Collection

Primary methods: direct personal interview, indirect oral investigation, questionnaire (mailed), schedules filled by enumerators, direct observation, and information from local agents/correspondents.

Secondary sources: published sources (government/CBS reports, journals, newspapers) and unpublished records (office files, research records).

Construction of a Frequency Distribution

1. Find the range $R = \text{max} - \text{min}$.
2. Decide the number of classes $k$ (often by Sturges' rule $k \approx 1 + 3.322\log_{10} N$).
3. Compute class width $h \approx R/k$ (rounded up).
4. Form class intervals (exclusive or inclusive) and choose limits.
5. Tally each observation into its class using tally marks.
6. Count tallies to get the frequency $f$ of each class.

The result is a table of class intervals against frequencies; cumulative frequencies (less-than / more-than) may be added.

Graphical Representation

• Histogram — adjacent rectangles, class intervals on the X-axis and frequency on the Y-axis; the area of each bar is proportional to frequency. Used for continuous data; bars touch each other. The mode can be located graphically from the tallest bar.
• Frequency polygon — a line graph obtained by plotting frequencies against class mid-points and joining them by straight lines; the ends are joined to the X-axis at the mid-points of the adjacent empty classes. It can be drawn over the histogram by joining the tops of the bars.
• Ogive (cumulative frequency curve) — a smooth curve obtained by plotting cumulative frequencies against class boundaries. The less-than ogive rises to the right; the more-than ogive falls to the right. Their intersection gives the median; quartiles can also be read off.

Skewness

Skewness measures the lack of symmetry of a distribution. A symmetric distribution has skewness $0$; a distribution with a longer right tail is positively skewed and one with a longer left tail is negatively skewed.

Shape relations:

• Symmetric: $\text{Mean} = \text{Median} = \text{Mode}$.
• Positive skew: $\text{Mean} > \text{Median} > \text{Mode}$.
• Negative skew: $\text{Mean} < \text{Median} < \text{Mode}$.

Kurtosis

Kurtosis measures the peakedness (and tail weight) of a distribution relative to the normal curve. Using $\beta_2 = \mu_4/\mu_2^2$:

• $\beta_2 = 3$: mesokurtic (normal).
• $\beta_2 > 3$: leptokurtic (more peaked, heavy tails).
• $\beta_2 < 3$: platykurtic (flatter).

Coefficients of Skewness

Karl Pearson's coefficient:

Bowley's (quartile) coefficient:

Worked example

For a distribution with Mean = 50, Mode = 44, $\sigma$ = 12:

For $Q_1 = 30,\; Q_2 = 40,\; Q_3 = 55$ (Bowley):

(Substitute the actual values given in the paper; the method is identical.)

Binomial Distribution

A discrete random variable $X$ follows a binomial distribution if it counts the number of successes in $n$ independent trials, each with constant success probability $p$ (and $q = 1-p$). The probability mass function is:

Properties

• It is a discrete distribution with parameters $n$ and $p$.
• Trials are independent (Bernoulli trials) with two outcomes (success/failure).
• $p$ remains constant across trials.
• Mean $= np$, Variance $= npq$, so Variance $\le$ Mean (since $q \le 1$).
• Standard deviation $= \sqrt{npq}$.
• It is positively skewed if $p < 0.5$, symmetric if $p = 0.5$.

Numerical (coin tossed 6 times)

Here $n = 6$, $p = P(\text{head}) = \tfrac12$, $q = \tfrac12$. So

Exactly 4 heads:

At least 4 heads $= P(X=4) + P(X=5) + P(X=6)$:

Skewness is a measure of the degree of asymmetry (lack of symmetry) of a frequency distribution about its mean. If the longer tail lies to the right it is positively skewed ($\text{Mean} > \text{Mode}$); if to the left it is negatively skewed ($\text{Mean} < \text{Mode}$); a symmetric distribution has zero skewness. A common measure is Karl Pearson's coefficient $S_k = \dfrac{\text{Mean}-\text{Mode}}{\sigma}$.

Kurtosis measures the peakedness or flatness of a frequency distribution compared with the normal distribution. It is based on the fourth moment, $\beta_2 = \dfrac{\mu_4}{\mu_2^{2}}$. A distribution is mesokurtic (normal) when $\beta_2 = 3$, leptokurtic (more peaked, heavy tails) when $\beta_2 > 3$, and platykurtic (flatter) when $\beta_2 < 3$.

Properties of the binomial distribution $B(n,p)$ with $q=1-p$:

• It is a discrete probability distribution with two parameters $n$ and $p$.
• Each of the $n$ trials is independent with only two outcomes (success/failure) and constant success probability $p$.
• pmf: $P(X=x) = \binom{n}{x}p^x q^{\,n-x}$.
• Mean $= np$ and Variance $= npq$; hence variance is always less than the mean.
• Standard deviation $= \sqrt{npq}$.
• The distribution is symmetric when $p = 0.5$, positively skewed when $p < 0.5$, and negatively skewed when $p > 0.5$.

Quartiles are the three values that divide an ordered data set into four equal parts, each containing 25% of the observations.

• First quartile $Q_1$ (lower quartile) — 25% of the data lie below it.
• Second quartile $Q_2$ — the median, with 50% below it.
• Third quartile $Q_3$ (upper quartile) — 75% of the data lie below it.

For grouped data, $Q_i = L + \dfrac{\frac{iN}{4} - C}{f}\times h$, where $L$ is the lower boundary of the quartile class, $C$ the preceding cumulative frequency, $f$ the class frequency and $h$ the class width.

The geometric mean (GM) of $n$ positive observations is the $n$-th root of their product:

In computational form, $GM = \text{antilog}\!\left(\dfrac{\sum \log x_i}{n}\right)$. It is appropriate for averaging ratios, rates of growth, and index numbers, and it is unduly influenced if any value is zero or negative.

Example: GM of $2, 4, 8 = \sqrt[3]{2\cdot4\cdot8} = \sqrt[3]{64} = 4$.

Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the other, i.e.

Equivalently $P(A\mid B) = P(A)$.

Example: Tossing a fair coin twice. The outcome of the first toss (say a head) does not influence the second toss, so

An ogive is the graph of a cumulative frequency distribution — a smooth curve obtained by plotting cumulative frequencies (Y-axis) against the upper or lower class boundaries (X-axis).

• Less-than ogive: plot less-than cumulative frequencies against upper boundaries; the curve rises to the right.
• More-than ogive: plot more-than cumulative frequencies against lower boundaries; the curve falls to the right.

The abscissa of the point where the two ogives intersect gives the median; quartiles, deciles and percentiles can also be read from an ogive.

The first 10 natural numbers are $1,2,3,\dots,10$.

Discrete variable — a quantitative variable that can take only isolated, countable values (usually whole numbers), with gaps between possible values. Example: number of students in a class, number of cars (you cannot have 2.5 cars).

Continuous variable — a quantitative variable that can take any value within a given range (including fractions and decimals), limited only by measuring precision. Example: height, weight, temperature, or time.

In short, discrete data are obtained by counting while continuous data are obtained by measuring.