Basic Statistics and Data Analysis

Statistics Lecture Notes, Online MCQs

Category: Miscellaneous Articles

Miscellaneous Articles

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Estimation, Approximating a Precise Value

Estimation (Approximating a Precise Value) is very useful especially when someone wishes to know whether he/ she has arrived at a logical solution to a problem under study. It is useful to learn about how to estimate the total sum of a bill to avoid immediate overpayments. For example, one can estimate the total amount of shop (supermarket) receipts. The estimate of these receipts can be done by rounding the amount of each item to the nearest half and keeping a running total mentally from the first item to the last one.

Suppose the following is a shop receipt, with the estimated amount and running total. Consider, the estimation, approximating a precise value for a utility bill.
Shop Item, Actual Amount, Estimated Amount, Running Total.

Shop ItemActual AmountEstimated AmountRunning Total
Item 14.504.504.50
Item 23.503.508
Item 31.31.59.5
Item 40.600.510
Item 52.95313
Item 62.85316
Item 71.601.5017.5
Item 82.75320.5
Item 92.42.523
Total22.4523 

From the above canadian online pharmacy example, it can be observed that estimation is a process of finding an estimate of a value. It saves time and results in the nearest possible exact value. An estimate can be overestimated (when the estimate exceeded the actual value) and underestimate (when the estimate fell short of the actual value).

Estimation, Approximating a Precise Value

In some cases, an estimate can be performed to round all of the numbers that you are working to the nearest 10 (or 100 or 1000) and then do the necessary calculations. In everyday life, the estimation can be used before you do a problem in an easier and faster way. It helps you to determine whether your answer is reasonable. Estimation is also useful when you need an approximate amount instead of a precise value.

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Absolute Error of Measurement

Absolute error of a measurement is the difference between the measured value of an object and its true value.

When we take the measurement of an object, it is possible that the measured value is either a little more or a little lower than its true value, that is, an absolute error has occurred.
For example, if a scale (a measurement device) states the weight 10KG but you know the true weight is about 9KG, then the scale has an absolute error of 1KG ($ 10KG-9KG=1KG$).

This error may be caused by the scale used itself ‌ not measuring the exact amount of measurement you are trying to measure. For example, your measuring device may be accurate to the nearest KG. That is, if the weight is 9.6KG, your scale may “round up” and gave 10KG. Thus, the absolute error is about $ 10KG-9.6KG=0.4KG$.

Mathematically, it can be described by the formula given below,
$ (\Delta X)=X_i-X$, where $ X_i$ is the measurement quantity by the device used and $X$ is the true value.

It is possible that the measurement device may either little more or a little lower than the true value, the formula can be described in absolute form, that is
$$(\Delta X)=|X_i-X| $$

Absolute Error of Measurement

‌Note that

  • If someones know the true value and the measured value, then the absolute error of measurement is just the subtraction of these numbers. However, sometimes, one may not know about the true value, one should use the maximum possible error as the absolute error.
  • Any possible measurement that one makes is ‌ an approximation, 100% accuracy of any measurement is impossible. It is also possible that if a measurement is of the same object is made twice, then the two measurements may not be identical. Such ‌ differences between measurements (of the same object) are called a variation in the measurement.
  • The absolute error of measurement does not provide any details about the graveness or importance of the error. For example, when measuring the distances between cities Kilometers apart ‌, an error of few centimeters is negligible. However, an error of centimeters when measuring a small piece of a machine is a ‌ significant error.
  • The largest possible absolute error of a measurement is always half of the value of the smallest unit used.

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Basics of Ratio: Use and Application

This post will discuss some Basics of the Ratio. The ratio is used to compare two quantities of the same kind. Consider in a group of 45 people, 15 of them are females. Let us understand the Basics of the Ratio from an example. We can compare the number of males and the number of females in the group in two different ways,

  1. There are 15 more males than females in a group of people. Actually, we are comparing the number of males and the number of females in the group of people by finding their difference.
  2. The number of males in a group of people is twice that of females. Actually, we are comparing the number of males and the number of females by finding a fraction consisting of the number of males over the number of females. The fraction is $\frac{30}{15}$.

In the second method, the fraction obtained is an example of the ratio.

A ratio may be written with a colon symbol between the numbers. Therefore, the male-female ratio in the group of people can be written as 30:15 or $\frac{30}{15}$. It can be read as the ratio of 30 is to 15 or simply 30 to 15.

In general, the ratio of $a$ to $b$ can be written as $la:b$ or $\frac{a}{b}$, where $a$ and $b$ represent whole numbers and $b$ should not be zero. Since ratio $\frac{30}{15}=\frac{2}{1}$ or $30:15 = 2:1$. the $30:15$ and $2:1$ are called equivalent ratios.

Note that the order in which the ratio is written is important. A ratio has no measurement units. It is only a number that indicates how many times, one quantity is as great as the other. For example, the male to the female ratio of 2:1 means that the number of males is twice the number of females. It can also be interpreted as the female to male of 1:2 or $\frac{1}{2}$ indicates that there are half as many females as males.

Ratios can also be used to compared more than two quantities. For example, three-man A, B, and C share the profit of a business. They receive say 5000, 3000 and 1000, respectively. The ratio of their share of the profit is 5000:3000:1000 or 5:3:1.

One can also find that either there is an increase or decrease in the ratio. For example, say if the number of teachers in a college is increased from 45 to 55, then the ratio “number of present staff” to “number of old staff” (or number of present staff: number of old staff) = 55:45 = 11:9 or $\frac{\text{no. of present staff}}{\text{no. of old staff}} = \frac{55}{45}=\frac{11}{9}$.
This ratio can be interpreted as the number of teachers has been increased in the ratio 11:9 or $\frac{11}{9}$.

Ratio

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Significant Figures: Introduction and Example

Rounding of numbers is done so that one can concentrate on the most important or significant digits. For example, consider a flat price at 285500. A rich man might think in hundreds of thousands of dollars. To a rich man, it is easier to think in terms of 1 significant figure, the “3” in 300,000. A wage might e worried about the hundreds of dollars. To him, there may be four significant figures, the ‘2’, ‘8’, ‘5’, ‘5’ in 285500.

Consider an example: A wight recorded as 8426kg is correct to 3 decimal places. Reporting this weight in grams, the 8425g is nearest to the whole number. Recording the weight as 8.426kg correct to 4 significant figures and converting the weight to 8426g, the number of significant figures is still 4. Thus, sometimes it is more useful to express a result in terms of numbers of significant figures rather than the number of decimal places.

There are some rules for writing significant figures:

Rule 1: Include one extra figure for consideration. Simply drop the extra figure if it is less than 5. If it is 5 or more, add 1 to the previous figure before dropping the extra figure.

Rule 2: All non-zero digits are significant wherever they are recorded. For example, 7.22 is correct to 3 significant figures.

Rule 3: Zeros that lies between non-zero digits are significant. For example, 2003 is correct to 4 significant figures.
Rule 4: Zeros which are not preceded by a non-zero digit (leading zeros) are not significant. For example, 0.000325 is correct to 3 significant digits.

Rule 5: Zeros that appear after the decimal points (trailing zeros) but are not followed by a non-zero digit are significant. For example, there are 5 significant digits in 22.300.

Rule 6: The final zeros in a whole number may or may not be significant. It depends on how the estimation is made.

A point to remember is that the number of digits is used to denote an exact value to a specified degree of accuracy. For example, 6084.324 is a value accurate to 7 significant figures. If written as 6080 it is accurate to 3 significant digits. The final 0 is not significant because it is used to show the order of magnitude of the number.

Significant Figures