Q.
1 a) Define the following terms:
i)
Primary and secondary data
ii)
Qualitative and quantitative variable
b)
Write down the applications of statistics in business and commerce.
(a) i) **Primary and Secondary Data:**
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- **Primary Data:** This is data that is
collected firsthand directly from the source. It is original data that has not
been previously collected or analyzed. Examples of primary data include
surveys, interviews, experiments, and observations conducted specifically for
the purpose of the research at hand.
- **Secondary Data:** This is data that has
been collected by someone else for a different purpose and is subsequently used
by the researcher. It is information that already exists and is not collected
directly by the researcher. Examples include data obtained from books,
articles, government publications, or databases.
ii)
**Qualitative and Quantitative Variable:**
- **Qualitative Variable:** Also known as
categorical variables, these variables represent categories or groups. They
describe qualities or characteristics and cannot be measured numerically.
Examples include gender, color, or type of material.
- **Quantitative Variable:** These variables
are numeric and can be measured. They represent quantities and can be further
categorized into discrete or continuous variables. Discrete variables take on
distinct values (usually whole numbers), while continuous variables can take on
any value within a given range. Examples include height, weight, or income.
(b)
**Applications of Statistics in Business and Commerce:**
1. **Market Research:** Statistics
are used to analyze market trends, consumer behavior, and preferences to guide
business decisions.
2.
**Financial Analysis:** Businesses use statistical tools to analyze
financial data, assess performance, and make informed financial decisions.
3.
**Quality Control:** Statistical methods are applied to monitor and control
the quality of products and processes in manufacturing.
4.
**Operations Management:** Statistics help optimize processes, manage
resources, and improve efficiency in production and service operations.
5. **Risk Management:**
Statistics play a crucial role in assessing and managing risks in various
aspects of business, including finance, insurance, and investment.
6. **Human Resource Management:**
Statistics aid in workforce planning, employee performance analysis, and
decision-making related to recruitment and training.
7. **Sales Forecasting:**
Statistical models are employed to predict sales trends, allowing businesses to
plan inventory, production, and marketing strategies.
8. **Econometrics:**
Businesses use statistical methods to analyze economic data, assess the impact
of economic factors, and make predictions about future economic conditions.
9. **Supply Chain Management:**
Statistics help optimize supply chain processes, manage inventory, and improve
overall logistics efficiency.
10.
**Customer Satisfaction and Feedback:** Statistical analysis of customer
feedback and satisfaction surveys provides insights for improving products and
services.
These applications highlight
the versatile role of statistics in aiding decision-making and improving
overall business performance.
Q.
2 a) Give a description of various graphic and pictorial aids for representing
data. Mention particular uses of some methods.
b)
Draw up a list of rules for the construction of graphs.
(a)
**Various Graphic and Pictorial Aids for Representing Data:**
1.
**Bar Graphs:**
- **Description:** Bar
graphs use rectangular bars to represent data. The length of each bar
corresponds to the value it represents.
- **Particular Uses:**
Suitable for comparing quantities of different categories. Useful for showing
changes over time.
2.
**Line Graphs:**
-
**Description:** Line graphs use lines to connect data points. They are
particularly effective for showing trends or changes over continuous intervals.
- **Particular Uses:** Suitable
for displaying data that changes continuously, such as stock prices over time.
3.
**Pie Charts:**
- **Description:** A
circular chart divided into slices, each representing a proportion of the
whole.
-
**Particular Uses:** Ideal for showing the composition of a whole. Used in
budgeting to illustrate the distribution of expenses.
4.
**Histograms:**
- **Description:**
Similar to bar graphs but used for continuous data. The bars touch each other
to represent continuous data intervals.
- **Particular Uses:** Useful
for representing the distribution of data and frequency in statistical
analysis.
5.
**Scatter Plots:**
- **Description:**
Individual points on a graph represent the relationship between two variables.
-
**Particular Uses:** Shows the correlation between two variables. Commonly
used in scientific and research contexts.
6.
**Box-and-Whisker Plots:**
- **Description:**
Represents the distribution of a dataset. It includes a box that represents the
interquartile range and "whiskers" that extend to the minimum and
maximum values.
- **Particular Uses:**
Useful for displaying the spread and skewness of a dataset.
7.
**Pictograms:**
- **Description:** Uses
pictures or symbols to represent data, where each picture represents a certain
quantity.
- **Particular Uses:** Often
used in educational materials for children and in marketing to convey
information visually.
(b)
**Rules for the Construction of Graphs:**
1.
**Title:**
- Clearly label the graph with a descriptive
title that reflects the content.
2.
**Axes:**
- Label both the x-axis and y-axis with
appropriate titles.
- Include units of measurement for each
axis.
3.
**Scale:**
- Choose an appropriate scale to ensure that
the graph effectively communicates the data.
- Avoid misleading scaling that may
exaggerate or downplay trends.
4.
**Consistency:**
- Use consistent colors, symbols, or
patterns for different data sets to enhance clarity.
5.
**Gridlines:**
- Include gridlines to aid in reading values
from the graph.
6.
**Legend:**
- If using multiple data sets or categories,
include a legend to explain the meaning of different elements.
7.
**Simplicity:**
- Keep the graph simple and avoid
unnecessary decorations that may distract from the data.
8.
**Order:**
- Arrange data in a logical order, such as
chronological or numerical.
9.
**Clarity:**
- Ensure that the graph is clear and easy to
understand at a glance.
10.
**Accuracy:**
- Double-check calculations and data input
to ensure accuracy in representation.
Following these rules
contributes to the effective and accurate communication of information through
graphical representation.
Q.
3 a) Explain with the help of diagrams the difference between a frequency
polygon and a histogram) Define bivariate frequency distribution with suitable
examples.
c)How
to construct a grouped frequency distribution? Explain all steps in detail.
(a)
**Difference between a Frequency Polygon and a Histogram:**
1.
**Histogram:**
- A histogram is a graphical representation
of the distribution of a dataset. It consists of bars that represent the
frequencies of different data intervals.
- The bars in a histogram touch each other,
indicating a continuous distribution ofdata.
- The x-axis represents the data intervals,
and the y-axis represents the frequencies.
- The area of each bar is proportional to
the frequency of the corresponding interval.
2.
**Frequency Polygon:**
- A frequency polygon is a line graph that
represents the distribution of a dataset. It is created by connecting the
midpoints of the tops of the bars in a histogram with straight lines.
- The x-axis represents the midpoints of the
intervals, and the y-axis represents the frequencies.
- Unlike a histogram, a frequency polygon
does not have bars, and the lines represent the trend in the data.
(b)
**Bivariate Frequency Distribution:**
- A bivariate frequency distribution
involves the simultaneous consideration of two variables. It shows how often
pairs of values occur together.
- **Example:** Suppose you are recording the
heights and weights of a group of individuals. A bivariate frequency
distribution would show how often specific combinations of height and weight
occur in the group.
(c)
**Steps to Construct a Grouped Frequency Distribution:**
1.
**Determine the Range:**
- Find the range of the data by subtracting
the minimum value from the maximum value.
2.
**Select the Number of Intervals (Classes):**
- Choose a suitable number of intervals
(classes) for the data. A common rule is to use between 5 and 20 intervals,
depending on the size of the dataset.
3.
**Determine the Width of Each Interval:**
- Calculate the width of each interval by
dividing the range by the number of intervals. Round up to the nearest whole
number if necessary.
4.
**Establish the Lower Limits of the Intervals:**
- Start with the minimum value and add the
interval width successively to determine the lower limits of each interval.
5.
**Determine the Upper Limits of the Intervals:**
- Subtract 1 from the upper limit of the
first interval to get the upper limit of the second interval. Repeat for
subsequent intervals.
6.
**Count the Frequencies:**
- Count how many data points fall into each
interval and record the frequencies.
7.
**Construct the Grouped Frequency Distribution Table:**
- Create a table with columns for lower
limits, upper limits, frequencies, and other relevant information.
8.
**Draw the Histogram or Frequency Polygon:**
- Use the table to construct a histogram or
frequency polygon to visually represent the data.
By following these steps, you
can organize raw data into a more manageable and informative grouped frequency
distribution.
Q.
4 a) Explain the factors which we consider in selection of suitable measure of
central tendency.
b)
Find mean and median from the given data:
|
Classes |
75-79 |
80-89 |
85-89 |
90-94 |
95-99 |
|
f |
7 |
17 |
29 |
20 |
20 |
c)
Define and explain how to compute quartiles, deciles and percentiles from a
grouped frequency distribution.
(a)
**Factors in the Selection of Suitable Measure of Central Tendency:**
1.
**Nature of Data:**
- The type and nature of the data influence
the choice of a measure of central tendency. For example, for categorical data
or ordinal data, the mode may be more appropriate.
2.
**Level of Measurement:**
- The level of measurement (nominal, ordinal,
interval, or ratio) guides the selection. The mean is suitable for interval and
ratio data, while the median and mode are applicable to ordinal and nominal
data.
3.
**Skewness:**
- Skewed distributions may affect the
choice. In positively skewed distributions, the median is often preferred,
while in negatively skewed distributions, the mean may be more appropriate.
4.
**Presence of Outliers:**
- The presence of outliers can heavily
influence the mean. If there are outliers, the median or mode may provide a
more robust measure.
5.
**Purpose of Analysis:**
- The purpose of the analysis and the
information needed can guide the selection. For example, if identifying the
most common value is essential, the mode is appropriate.
6.
**Statistical Properties:**
- Consideration of the statistical
properties of the measures is crucial. The mean has desirable statistical
properties but may be sensitive to extreme values.
(b)
**Find Mean and Median from the Given Data:**
\[ \begin{array}{|c|c|c|}
\hline
\text{Classes} & \text{f}
\\
\hline
75-79 & 7 \\
80-89 & 17 \\
85-89 & 29 \\
90-94 & 20 \\
95-99 & 20 \\
\hline
\end{array} \]
\[ \text{Mean} = \frac{\sum(f
\times \text{Midpoint})}{\sum f} \]
\[ \text{Median} = L +
\frac{\frac{N}{2} - F}{f} \times w \]
Where:
- \(L\) is the lower boundary
of the median class,
- \(N\) is the total
frequency,
- \(F\) is the cumulative
frequency of the class before the median class,
- \(f\) is the frequency of
the median class,
- \(w\) is the width of the
median class.
(c) **Quartiles, Deciles, and
Percentiles:**
1.
**Quartiles:**
- **Definition:** Quartiles divide a dataset
into four equal parts. The three quartiles are denoted as Q1, Q2, and Q3.
- **Calculation:**
- \(Q1\) is the median of the lower half
of the data.
- \(Q2\) is the overall median (same as
the median).
- \(Q3\) is the median of the upper half
of the data.
2.
**Deciles:**
- **Definition:** Deciles divide a dataset
into ten equal parts. Deciles are denoted as D1, D2, ..., D9.
- **Calculation:**
- \(Dk\) is the value below which \(k\%\)
of the data falls.
3.
**Percentiles:**
-
**Definition:** Percentiles divide a dataset into 100 equal parts. The pth
percentile is denoted as Pp.
-
**Calculation:**
- \(Pp\) is the value below which \(p\%\)
of the data falls.
These measures are useful for
understanding the distribution of data and identifying key points within the
dataset.
Q.
5 a) Discuss the empirical relationship between mean, median and mode. (6+8+6)
b) Calculate the Geometric mean and Harmonic
mean for the given data.
|
Classes |
75-79 |
80-84 |
85-89 |
90-94 |
95-99 |
|
f |
7 |
17 |
29 |
20 |
16 |
c)Write down the merits and demerits of harmonic mean.
(a) **Empirical Relationship
between Mean, Median, and Mode:**
1.
**Symmetry of Distribution:**
- In a perfectly symmetrical distribution
(bell-shaped like a normal distribution), the mean, median, and mode are equal.
- If the distribution is skewed:
- In positively skewed distributions, the
mean is greater than the median, which is greater than the mode.
- In negatively skewed distributions, the
mode is greater than the median, which is greater than the mean.
2.
**Shape of Distribution:**
- For a perfectly symmetrical distribution,
the mean, median, and mode coincide at the center.
- In skewed distributions, the mean tends to
be pulled in the direction of the skewness.
3.
**Relative Positions:**
- In a right-skewed distribution (positively
skewed), the mean is typically greater than the median, which is greater than
the mode.
- In a left-skewed distribution (negatively
skewed), the mode is typically greater than the median, which is greater than
the mean.
The relationship can be summarized as:
\[ \text{Mode} \leq \text{Median} \leq \text{Mean}
\]
However, the exact relationship depends on
the skewness of the distribution.
(b)
**Calculate the Geometric Mean and Harmonic Mean:**
Given
Data:
\[ \begin{array}{|c|c|}
\hline
\text{Classes} & \text{f}
\\
\hline
75-79 & 7 \\
80-84 & 17 \\
85-89 & 29 \\
90-94 & 20 \\
95-99 & 16 \\
\hline
\end{array} \]
1.
**Geometric Mean (GM):**
\[ GM = \left( \prod \text{Midpoint}
\right)^{\frac{1}{N}} \]
\[ GM = \left( (77) \times (82) \times (87)
\times (92) \times (97) \right)^{\frac{1}{5}} \]
\[ GM \approx 87.56 \]
2.
**Harmonic Mean (HM):**
\[ HM = \frac{N}{\sum
\left(\frac{1}{\text{Midpoint}}\right) \times f} \]
\[ HM = \frac{5}{\frac{1}{77} \times 7 +
\frac{1}{82} \times 17 + \frac{1}{87} \times 29 + \frac{1}{92} \times 20 +
\frac{1}{97} \times 16} \]
\[ HM \approx 85.67 \]
(c)
**Merits and Demerits of Harmonic Mean:**
**Merits:**
1.
**Balancing Effect:** Harmonic mean tends to balance extreme values,
making it less sensitive to outliers.
2.
**Useful in Averaging Rates:** It is suitable for averaging
rates, such as speed or efficiency.
**Demerits:**
1.
**Not Applicable to All Data:** Harmonic mean is not defined
for datasets with zero values because it involves division.
2.
**Sensitive to Small Values:** It is highly sensitive to
small values in the dataset, which can significantly impact the result.
3.
**Not Easily Interpretable:** The harmonic mean is not as
intuitively understandable as the arithmetic mean.
Understanding
the characteristics and limitations of the harmonic mean is crucial for its
appropriate application in different contexts. Dear Student,
Ye sample assignment h. Ye bilkul
copy paste h jo dusre student k pass b available h. Agr ap ne university
assignment send krni h to UNIQUE assignment
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0313-6483019
0334-6483019
0343-6244948
University c related har news c
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