2.9 Descriptive statistics: measuring the spread of the data (Page 3/7)

Collaborative statistics using Page 3 / 7

In these formulas, $f$ represents the frequency with which a value appears. For example, if a value appears once, $f$ is 1. If a value appears three times in the data set or population, $f$ is 3.

Sampling variability of a statistic

The statistic of a sampling distribution was discussed in Descriptive Statistics: Measuring the Center of the Data . How much the statistic varies from one sample to another is known as the sampling variability of a statistic . You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example of a standard error. It is a special standard deviation and is known as the standard deviation of the sampling distribution of the mean. You will cover the standard error of the mean in The Central Limit Theorem (not now). The notation for the standard error of the mean is

\frac{σ}{\sqrt{n}}

where

σ

is the standard deviation of the population and

n

is the size of the sample.

In practice, USE A CALCULATOR OR COMPUTER SOFTWARE TO CALCULATE THE STANDARD DEVIATION. If you are using a TI-83,83+,84+ calculator, you need to select the appropriate standard deviation $σ_{x}$ or $s_{x}$ from the summary statistics. We will concentrate on using and interpreting the information that the standard deviation gives us. However you should study the following step-by-step example to help you understand how the standard deviation measures variation from the mean.

In a fifth grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students. The following data are the ages for a SAMPLE of $n = 20$ fifth grade students. The ages are rounded to the nearest half year:

9
9.5
9.5
10
10
10
10
10.5
10.5
10.5
10.5
11
11
11
11
11
11
11.5
11.5
11.5

\bar{x} = \frac{9 + 9.5 \times 2 + 10 \times 4 + 10.5 \times 4 + 11 \times 6 + 11.5 \times 3}{20} = 10.525

The average age is 10.53 years, rounded to 2 places.

The variance may be calculated by using a table. Then the standard deviation is calculated by taking the square root of the variance. We will explain the parts of the table after calculating $s$ .

Data	Freq.	Deviations	${Deviations}^{2}$	(Freq.)( ${Deviations}^{2}$ )
$x$	$f$	$(x - x)$	${(x - \bar{x})}^{2}$	$(f) {(x - \bar{x})}^{2}$
$9$	$1$	$9 - 10.525 = - 1.525$	${(- 1.525)}^{2} = 2.325625$	$1 \times 2.325625 = 2.325625$
$9.5$	$2$	$9.5 - 10.525 = - 1.025$	${(- 1.025)}^{2} = 1.050625$	$2 \times 1.050625 = 2.101250$
$10$	$4$	$10 - 10.525 = - 0.525$	${(- 0.525)}^{2} = 0.275625$	$4 \times .275625 = 1.1025$
$10.5$	$4$	$10.5 - 10.525 = - 0.025$	${(- 0.025)}^{2} = 0.000625$	$4 \times .000625 = .0025$
$11$	$6$	$11 - 10.525 = 0.475$	${(0.475)}^{2} = 0.225625$	$6 \times .225625 = 1.35375$
$11.5$	$3$	$11.5 - 10.525 = 0.975$	${(0.975)}^{2} = 0.950625$	$3 \times .950625 = 2.851875$

The sample variance, $s^{2}$ , is equal to the sum of the last column (9.7375) divided by the total number of data values minus one (20 - 1):

$s^{2} = \frac{9.7375}{20 - 1} = 0.5125$

The sample standard deviation $s$ is equal to the square root of the sample variance:

$s = \sqrt{0.5125} = . 0715891$ Rounded to two decimal places, $s = 0.72$

Typically, you do the calculation for the standard deviation on your calculator or computer . The intermediate results are not rounded. This is done for accuracy.

Verify the mean and standard deviation calculated above on your calculator or computer.

Using the ti-83,83+,84+ calculators

Enter data into the list editor. Press STAT 1:EDIT. If necessary, clear the lists by arrowing up into the name. Press CLEAR and arrow down.
Put the data values (9, 9.5, 10, 10.5, 11, 11.5) into list L1 and the frequencies (1, 2, 4, 4, 6, 3) into list L2. Use the arrow keys to move around.
Press STAT and arrow to CALC. Press 1:1-VarStats and enter L1 (2nd 1), L2 (2nd 2). Do not forget the comma. Press ENTER.
$x$ =10.525
Use Sx because this is sample data (not a population): $Sx$ =0.715891

Questions & Answers

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In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

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other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

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what is monopoly mean?

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What is different between quantity demand and demand?

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Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

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Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

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it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

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In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

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Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

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Answer

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Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

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suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

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types of unemployment

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What is the difference between perfect competition and monopolistic competition?

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Source: OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23

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