How to Find the Area Right of a Z Score & Read a Z Table

Given any data value, we can place how far that information value is abroad from the hateful, just by doing a subtraction ten – μ . This value will exist positive if your information value lies above (to the right) of the mean, and negative if it lies below (to the left) of the mean. Only what nosotros'd actually like to know is, relative to the spread of our information set up, how far is x from μ ? Remember that the standard deviation σ gives us a mensurate of how spread out our entire set of individual information values is.

The z-score for any single data value can be found past the formula (in English):

or with symbols (equally seen before!):

Obviously a z-score will be positive if the information value lies above (to the right) of the mean, and negative if the data value lies beneath (to the left) of the mean.

Example 6.i: Calculating and Graphing z-Values

Given a normal distribution with μ = 48 and s = 5, convert an x-value of 45 to a z-value and point where this z-value would be on the standard normal distribution.

Solution

Brainstorm past finding the z-score for ten = 45 as follows.

Now describe each of the distributions, mark a standard score of z = −0.sixty on the standard normal distribution.

The distribution on the left is a normal distribution with a hateful of 48 and a standard deviation of 5. The distribution on the right is a standard normal distribution with a standard score of z = −0.sixty indicated.

Z -scores mensurate the distance of any data indicate from the mean in units of standard deviations and are useful because they allow usa to compare the relative positions of information values in different samples. In other words, the z-score allows us to standardize two or more normal distributions, or more appropriately, to put them on the same calibration. Therefore, nosotros'll be able to compare relative positions of data values within their ain distribution to determine which data values are closer to or farther from the mean. A prime instance for this is to compare the test scores for two students, one who scored a 28 on the ACT (scores range from 1 – 36) and another who scored a 1280 on the Sabbatum (scores range from 400 – 1600). Who, relative to their associated examination, scored better?

Example

Suppose you are enrolled in 3 classes, statistics, biological science, and kayaking, and you just took the first test in each. You receive a grade of 82 on your statistics exam, where the hateful class was 74 and the standard deviation was 12. You lot receive a course of 72 on your biology exam, where the mean form was 65 and the standard deviation was ten. Finally, yous receive a grade of 91 on your kayaking exam, where the hateful grade was 88 and the standard difference was 6. Although your highest test score was 91 (kayaking), in which grade did y'all score the best, relative to the rest of the class ? Nosotros tin respond this using a z-score!

Your statistics test score was 0.67 standard deviations better than the class average; your biology score was 0.7 standard deviations meliorate than the class average; your kayaking score was merely 0.five standard deviations better than the class average.  Therefore, even though your actual score on the biology examination was the lowest of the iii exam scores, relative to the distribution of all class exam scores , your biology exam score was the highest relative grade.

Finding an Area (Proportion) Given a Specific Z-Value

To determine the surface area under the Due north (0, 1) curve for any data value that does not autumn exactly 1, two, or three standard deviations above or below the mean actually requires some calculus. Lucky for us, areas under the N(0, 1) curve can exist obtained in numerous other ways, including technology (TI-83/84, Excel) and a tabular array of values. Search the Internet for "standard normal tabular array" and yous'll discover hundreds of tables illustrating z-scores and their associated areas. The majority of these methods written report the area to the left of the specified z-score z, no matter where it lies. This comes from a calculus performance of integration, which finds an surface area from the offset of a distribution (i.e., the far left-tail) upwards to the z-score. Two images are provided.

In that location are three types of area calculations that you lot will be performing, each requiring slightly different work:

• For areas to the left of z: just employ the area provided past a table or technology.
• For areas to the right of z: because the total area under a density curve is 1 (100%), just calculate: 1 − expanse to the left of z ₀ .
• For areas betwixt 2 z-values, say z₀ andz₁  (where z ₀< z ₁): find the area to the left ofz _ane and subtract from information technology the area to the left of z ₀ .

Finding a Z-Value Given an Area

This is a slightly more challenging chore than calculating an area, considering you basically piece of work "backwards" from an algebraic standpoint. It's of import to realize that a Standard Normal Tabular array has two parts: (i) the acme and side margins, which form the tenths and hundredths of a z-score, and (2) the body of the table, which are all the area (probability) values. Besides, recollect that the Standard Normal Table only provides united states of america information on the expanse (probability) to the left of a z-score. A modest extract of Tabular array B from Appendix A is shown beneath.

Notice that the z-values given in the table are rounded to 2 decimal places. The first decimal place of each z-value is listed in the left column, with the 2d decimal place in the superlative row. Where the appropriate row and column intersect, we detect the amount of area nether the standard normal curve to the left of that particular z-value.

Example : Finding Surface area to the Left of a Positive z-Value Using a Cumulative Normal Table

Notice the surface area under the standard normal bend to the left of z = 1.37.

Solution

To read the table, we must break the given z-value (1.37) into two parts: ane containing the first decimal place (1.3) and the other containing the second decimal place (0.07). So, in Table B from Appendix A, look across the row labeled i.3 and down the cavalcade labeled 0.07. The row and column intersect at 0.9147. Thus, the area nether the standard normal curve to the left of z = 1.37 is 0.9147.

Using a TI-83/84 Plus calculator, nosotros tin can find a value of the area to the left of a z-score. To obtain the solution using a TI-83/84 Plus calculator, perform the following steps.

• Printing 2nd and then Vars to access the DISTR carte du jour.
• Choose option 2:normalcdf( .
• Enter lower bound, upper spring, µ , σ. Note If you want to find area nether the standard normal curve, as in this example, then you lot do not need to enter µ or σ.
• Since we are asked to find the area to the left of z, the lower bound is -∞. From the empirical rule we know that after about 3 standard deviations abroad from the mean we take accounted for nigh all of the data, and then for our lower bound we volition but utilise a very negative number.We cannot enter -∞ into the reckoner, and so we will enter a very small value for the lower endpoint, such as -10⁹⁹. This number appears as -1E99 when entered correctly into the calculator. To enter -1E99, press(-) 1 [2^nd][ , ]99. This appears on the screen equally normcdf(-1E99,i.37,0,1).

If we are given an area (or probability) value, we demand to first locate it in the torso of a table, then track our way up and to the left in order to piece together the z-score that relates to the specified surface area. Proceed in mind that yous may not find the exact area value in the body of the table…so just utilize the closest value you can find, and then identify the proper z-score.

I calculation that will be used frequently in the coming chapters is to place the ii z-scores that separate a specific area in the center of the standard normal distribution.

Example

Suppose we want to know which two z-scores split out the heart 95% of the data. From the empirical rule, we already know the z-scores that do this are ±2 (2 standard deviations on either side of the mean). In reality, it's not exactly ±2, but close enough for rough calculations.

To discover the exact two z-scores, we utilize the following logic: If the center portion is 95% = 0.95, then how much area lies outside of the eye (to the left and right)? A simple subtraction solves this! i – 0.95 = 0.05. The "exterior" area, 0.05, must be separate as between the ii tails (because of symmetry!). Therefore, dividing 0.05 by 2 gives united states an area of 0.025 in each tail .

Using a standard normal table "backwards," nosotros first wait through the body of the table to find an area closest to 0.025. The z-score corresponding to a left-tail area of 0.025 isz = −1.96. At present, therefore, the upper z-score will be z = 1.96, by the symmetry belongings of the standard normal distribution. You could also discover the upper z-score past looking upward the area/probability value 0.025 + 0.95 = 0.975 in the body of the table and finding the associated z-value. By the end of the class, you will exist extremely familiar with z-scores that define a central xc% (z = ± 1.645), 95% (z = ± i.96), and 99% (z = ± ii.576).

Case: Discover and translate the probability of a random Normal variable

Suppose you simply purchased a 2005 Honda Insight with automatic transmission. Using www.fueleconomy.gov you determine for the 2005 Honda Insights have hateful highway gas milage is 56 miles per gallon with a standard difference of 3.two. The distribution of this data has a bong-shape and is normal. You want to know the following:

a) How likely is it that your Honda Insight with automatic transition will get ameliorate than 60 miles per gallon on the highway.

b) How likely is it that your Honda Insight with automatic transition will get less than fifty miles per gallon on the highway.

c) How likely is it that your Honda Insight with aoutomatic transition volition get betwixt 52 and 62 miles per gallon on the highway.

Solution

This problem deals with data that is unremarkably distributed with mean 56 and standard deviation 3.2, i.e., .

(a)

In symbols, we are asked to calculate P(10 > sixty). Sketching a normal bend and shading the area corresponding to greater than 60, gives us the graph shown. In order to calculate the appropriate area in the upper (right) tail, we must first convert our data to the standard normal distribution. The z-score for x = 60 is:

This means that 60 is 1.25 standard deviations to a higher place the mean. Discover how lining the two normal curves up every bit shown illustrates how the 2 areas are the aforementioned: P(X > threescore) = P(Z > 1.25).

Using z = 1.25, we go to Table IV (or apply normcdf(1.25,1E99,0,1)) to find the expanse to the left of z = 1.25 is 0.8943. Since we demand the surface area to the right, nosotros simply accept one – 0.8943 = 0.1057.

Therefore, P(X > lx) = 0.1057 = 10.57%. There are a couple ways to interpret this answer:

• Of all the model year 2005 Honda Insight cars produced with an automatic transmission, 10.57% will get over threescore miles per gallon on the highway.
• If yous went to a car lot and purchased a new model year 2005 Honda Insight cars produced with an automatic transmission, there is a x.57% chance that your car will go over 60 miles per gallon on the highway.

(b)

In symbols, we are asked to summate P(X < 50). Sketching a normal curve N(56, 3.ii)and shading the area respective to less than 50, gives us the graph shown to the right.

In social club to summate the appropriate area in the lower (left) tail, nosotros must first catechumen our data to the standard normal distribution. The z-score for x = 50 is:

Thus, the value 50 MPG is 1.88 standard deviations below the mean. In symbols nosotros see:P(10 < 50) = P(Z < −1.88).

Using z = -one.88, nosotros go to Table IV (or employ normcdf(-1E99,-ane.88,0,ane)) to find the surface area to the left of z = -1.88 is 0.0301. Therefore, P(10 < 50) = 0.0301 = 3.01%. There are a couple ways to interpret this answer:

• Of all the model year 2005 Honda Insight cars produced with an automatic transmission, 3.01% will get less than 50 miles per gallon on the highway.
• If you went to a car lot and purchased a new model year 2005 Honda Insight cars produced with an automated transmission, there is a 3.01% take a chance that your auto volition get less than 50 miles per gallon on the highway.

(c)

In symbols, we are asked to calculate P(58 < 10 < 62). Sketching a normal curve Due north(56, iii.2)  and shading the area respective to greater than 58 but less than 62, gives us the graph shown. In order to calculate the appropriate expanse, nosotros must first convert both data to the standard normal distribution.

The z-score for 10= 58 is:

and thez-score forx = 62 is:

In terms of probability, nosotros tin now say: P(58 < X < 62) = P(0.63 < Z < 1.88).

Using z = 1.88, we go to Table Four (or use technology) to find the area to the left of z = one.88 is 0.9699. Now, we need to remove (subtract) the area left of z = 0.63, which is 0.7357. Therefore, P(58 < X < 62) = 0.9699 – 0.7357 = 0.2342, or 23.42%. There are a couple ways to interpret this respond:

• Of all the model year 2005 Honda Insight cars produced with an automatic manual, 23.42% will get betwixt 58 and 62 miles per gallon on the highway.
• If you went to a machine lot and purchased a new model year 2005 Honda Insight cars produced with an automatic manual, at that place is a 23.42% adventure that your car will get between 58 and 62 miles per gallon on the highway.

This calculation can be done with both normcdf(0.63,1.88,0,1) and normcdf(58,62,56,iii.2), which will be the same.

Observe the Value of a Random Variable Knowing a Probability Value

In these types of issues, we demand to piece of work "backwards." Starting with a specified probability, find the specified z-score, then work our fashion back to the random variable. The tables of standard normal values are non a "one-manner" tool! What practice we mean by that? So far y'all've started with a value for a random variable (like a gas mileage value in the previous problem), turned it into a z-score, and then looked upwards the associated probability value for that z-score. We tin utilize this table to piece of work backwards! We tin get-go with a known probability value in the body of a tabular array, identify the z-score corresponding to that area by moving your fingers to the associated row and column, the contrary the algebra transformation from a z-score to a random variable.

If this sounds disruptive, think back to the steps we took in the preceding case:

If, however, we are given an area/probability, and so to work our style back to the original data value, we must offset identify the appropriate z-score, then "un-standardize" the z-score to get in (finally!) back at the information value. How practice we algebraically "undo" the z-score? Easy…but solve for the data value X:

Multiply both sides past σ to remove it from the denominator on the left side:

Ten – μ = Z⋅σ

Finally, add the value of μ to both sides to isolate the value of the random variable Ten:

X = Z⋅σ + μ

Example: Finding the value of a normal random variable

Instead you want to know a gas mileage for a item probability. Detect what gas mileage for your 2005 Honda Insight will go amend gas mileage than 97% of all other 2005 Honda Insights with automatics transmission.

Solution

This trouble again deals with information that is normally distributed with mean 56 and standard departure 3.2, i.e., North(56, 3.2).

To notice the 97% percentile gas mileage, we need to find the specific miles per gallon X that separates the lesser 97% of all gas mileages from the top 3%. So for this problem nosotros are given a percentage/area. Sketching the normal curve gives the graph shown.

Using Tabular array IV, we observe 0.97 in the body of the table, and then place the z-score of one.88. Notice that the verbal area 0.97 is not in the table, simply the closest area of 0.9699 has the z-score of ane.88. At present nosotros un-standardize the z-score of 1.88. In English this ways we demand to identify the specific gas mileage that is 1.88 standard deviations above the mean of 56. Solving for X in the Z transform gives:

Therefore, if your 2005 Honda Insight cars with an automatic transmission gets 62 mpg, it gets better miles per gallon than 97% of all 2005 Honda Insight cars with an automated manual.

How to Find the Area Right of a Z Score & Read a Z Table

Source: https://mat117.wisconsin.edu/4-the-z-score/

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