Business Mathematics and Statistics · Class 12 Commerce
Ch 9Applied Statistics (Time Series, Index Numbers, Statistical Quality Control) — Class 12 Business Mathematics and Statistics, concept-first.
A time series is a set of numerical values (sales, prices, production, rainfall, population, and so on) recorded at successive, usually equal, intervals of time — daily, weekly, monthly, quarterly, or yearly.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Components of a Time Series
Every observed value in a time series can be thought of as the joint outcome of four separate forces acting together: a secular trend (the slow, long-run direction), a seasonal variation (a pattern that repeats every yea…
Most relevant Q&A
- State which component of a time series (secular trend, seasonal variation, cyclical variation, or irregular variation) best explains each of…Free
- The multiplicative model of the time series with the components $T, S, C$ and $I$ is : (a) $y = T + S \times C \times I$ (b) $y = T \times S…Preview
- The seasonal variation means the variations occurring within : (a) a month (b) some years (c) a week (d) a yearPreview
- A time series consists of __________ components. (a) four (b) two (c) three (d) fivePreview
- Define Secular trend.Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Time Series: Meaning and Components
A time series is a set of numerical values (sales, prices, production, rainfall, population, and so on) recorded at successive, usually equal, intervals of time — daily, weekly, monthly, quarterly, or…
Measuring Trend: Method of Moving Averages
The method of moving averages measures trend by smoothing out the short-term ups and downs in a series, so that the underlying long-term direction becomes visible.
Measuring Trend: Method of Least Squares
The method of least squares fits a mathematical trend line to a time series so that trend can be measured (and, importantly, extended to forecast future years) without losing values at either end the…
Meaning and Uses of Index Numbers
An index number is a statistical measure that expresses the average relative change in a variable (or a group of related variables, like the prices of a basket of goods) between one period — the base…
Unweighted Index Numbers: Simple Aggregate and Simple Average of Price Relatives
The simplest way to combine several commodities into one index number ignores how important each commodity is (how much of it is actually bought) and treats every commodity equally.
Weighted Aggregate Methods: Laspeyres' and Paasche's Price Index
Because a simple index treats a commodity bought in huge quantity the same as one bought only rarely, real index numbers are weighted by quantity, so that commodities the public actually spends more o…
Cost of Living Index (Consumer Price Index)
A Cost of Living Index, also called a Consumer Price Index (CPI), measures the average change in the prices of a fixed 'basket' of goods and services that a particular group of consumers (industrial w…
Statistical Quality Control: Meaning and Causes of Variation
Statistical Quality Control (SQC) is the use of statistical methods to monitor and control the quality of a manufacturing process while production is actually going on, rather than only inspecting fin…
Control Charts: The X-bar Chart and the R Chart
A control chart is a graph with a Central Line (CL), an Upper Control Limit (UCL), and a Lower Control Limit (LCL), on which sample statistics are plotted in the order they were taken.
Tolerance Limits and Control Limits
Students often confuse control limits with tolerance (or specification) limits, but the two answer completely different questions and come from completely different sources.
Exercises
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- Q3State which component of a time series (secular trend, seasonal variation, cyclical variation, or irregular variation) best explains each of…Free
- Q4Explain why a 3-yearly moving-average trend series always has fewer values than the original time series it was calculated from.Free
- Q6State any two uses of index numbers in business and economic analysis.Free
- Q8Explain, with reasoning, why Laspeyres' Price Index tends to overestimate a genuine rise in the cost of living, while Paasche's Price Index…Preview
- Q9What is a Cost of Living Index (Consumer Price Index)? State one practical use of it.Preview
- Q10Distinguish between chance (random) causes and assignable causes of variation in a manufacturing process, with one example of each.Preview
- Q12In a control chart, a sample mean is found to plot above the Upper Control Limit (UCL). What does this indicate, and what action should be t…Preview
- Q13Distinguish between control limits and tolerance (specification) limits.Preview
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 26 questionsHide questions26 questions
- Q1The multiplicative model of the time series with the components $T, S, C$ and $I$ is : (a) $y = T + S \times C \times I$ (b) $y = T \times S…Preview
- Q2How many causes of variation will affect the quality of a product ? (a) $1$ (b) $4$ (c) $3$ (d) $2$Preview
- Q3Fit a trend line by the method of semi-averages for the given data. | Year | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | | --- | --- |…Preview
- Q4Calculate four-yearly moving averages of number of students studying in a higher secondary school in a particular city from the following da…Preview
- Q5The value of 'b' in the trend line $y = a + bx$ is : (a) Either positive or negative (b) Always positive (c) Zero (d) Always negativePreview
- Q6Which of the following Index number satisfy the time reversal test ? (a) Paasche's Index number (b) Fisher Index number (c) Laspeyre's Index…Preview
- Q7State the uses of Index number.Preview
- Q8Construct the cost of living Index number for 2015 on the basis of 2012 from the following data using family budget method. | Commodity | Pr…Preview
- Q9Fisher's price index number is the ________ between Laspeyre's and Paasche's price index number. (a) Arithmetic mean (b) Geometric mean (c)…Preview
- Q10The seasonal variation means the variations occurring within : (a) a month (b) some years (c) a week (d) a yearPreview
- Q11For $\sum p_0q_0=1974$, $\sum p_1q_0=3140$, $\sum p_1q_1=2005$ find the Cost of Living Index by Aggregate Expenditure Method.Preview
- Q12Construct the cost of living index number for 2011 on the basis of 2007 from the given data using Family Budget method. | Commodities | Pric…Preview
- Q13(a) Compute (i) Laspeyre's (ii) Paasche's (iii) Fisher's Index numbers for the year 2010 from the following data. | Commodity | Price 2000 |…Preview
- Q14(a) Given below are the values of sample mean ($\bar{X}$) and the range (R) for ten samples of size 5 each. Draw mean chart and comment on t…Preview
- Q15A time series consists of __________ components. (a) four (b) two (c) three (d) fivePreview
- Q16The quantities that can be numerically measured can be plotted on a : (a) $\bar{x}$-chart (b) $p$-chart (c) $np$-chart (d) $c$-chartPreview
- Q17Define Secular trend.Preview
- Q18A machine drills hole in a pipe with a mean diameter of $0.532$ cm and a standard deviation of $0.002$ cm. Calculate the control limits for…Preview
- Q19A time series is a set of data recorded : (a) Weekly (b) Successive points of time (c) Periodically (d) all the abovePreview
- Q20__________ price index number satisfies both the Time Reversal and Factor Reversal test. (a) Paache's (b) Fisher's (c) Laspeyre's (d) None o…Preview
- Q21Fit a trend line by the method of freehand method for the given data. | Year | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | | ---…Preview
- Q22(a) Calculate the seasonal index for the monthly sales of a product using the method of simple averages. | Months | Year 2001 | 2002 | 2003…Preview
- Q23Most commonly used index number is : (a) Price index number (b) Volume index number (c) Simple index number (d) Value index numberPreview
- Q24The upper control limit for $\bar{X}$ chart is given by : (a) $\bar{\bar{X}} + A_2 \bar{R}$ (b) $\bar{X} + A_2 \bar{R}$ (c) $\bar{\bar{X}} +…Preview
- Q25The following figures relate to the profits of a commercial concern for 8 years. | Year | Profit (₹) | | --- | --- | | 1986 | 15,420 | | 198…Preview
- Q26Calculate the cost of living index number for the following data. | Commodities | Quantity 2005 | Price 2005 | Price 2010 | | --- | --- | --…Preview
More questions
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- Example 1A company's sales (₹ lakh) for the years 2015 to 2021 were as follows. Calculate the trend values by the method of 3-yearly moving averages.…Free
- Example 2Fit a straight-line trend by the method of least squares to the following data on a shop's sales, and estimate the trend value for 2022. | Y…Free
- Example 5From the following data on the prices of four commodities, calculate the price index for the current year using (a) the Simple Aggregate Met…Preview
- Example 7From the following price and quantity data for three commodities, calculate (a) Laspeyres' Price Index and (b) Paasche's Price Index. | Comm…Preview
- Example 11Five samples, each of size 5, were drawn from a production process. Their means and ranges were: | Sample | 1 | 2 | 3 | 4 | 5 | |---|---|---…Preview