# 3.2 Normal distribution: z-scores

 Page 1 / 1

If $X$ is a normally distributed random variable and $X$ ~ $\mathrm{N\left(\mu , \sigma \right)}$ , then the z-score is:

$z=\frac{x-\mu }{\sigma }$

The z-score tells you how many standard deviations that the value $x$ is above (to the right of) or below (to the left of) the mean, $\mu$ . Values of $x$ that are larger than the mean have positive z-scores and values of $x$ that are smaller than the mean have negative z-scores. If $x$ equals the mean, then $x$ has a z-score of $0$ .

Suppose $X$ ~ $\mathrm{N\left(5, 6\right)}$ . This says that $X$ is a normally distributed random variable with mean $\mathrm{\mu = 5}$ and standard deviation $\mathrm{\sigma = 6}$ . Suppose $\mathrm{x = 17}$ . Then:

$z=\frac{x-\mu }{\sigma }=\frac{17-5}{6}=2$

This means that $\mathrm{x = 17}$ is 2 standard deviations $\mathrm{\left(2\sigma \right)}$ above or to the right of the mean $\mathrm{\mu = 5}$ . The standard deviation is $\mathrm{\sigma = 6}$ .

Notice that:

$5+2\cdot 6=17\phantom{\rule{20pt}{0ex}}\text{(The pattern is}\phantom{\rule{5pt}{0ex}}\mu +z\sigma =x.\text{)}$

Now suppose $\mathrm{x=1}$ . Then:

$z=\frac{x-\mu }{\sigma }=\frac{1-5}{6}=-0.67\phantom{\rule{20pt}{0ex}}\text{(rounded to two decimal places)}$

This means that $\mathrm{x = 1}$ is 0.67 standard deviations $\mathrm{\left(- 0.67\sigma \right)}$ below or to the left of the mean $\mathrm{\mu = 5}$ . Notice that:

$5+\left(-0.67\right)\left(6\right)$ is approximately equal to 1 $\phantom{\rule{20pt}{0ex}}$ (This has the pattern $\mu +\left(-0.67\right)\sigma =1$ )

Summarizing, when $z$ is positive, $x$ is above or to the right of $\mu$ and when $z$ is negative, $x$ is to the left of or below $\mu$ .

Some doctors believe that a person can lose 5 pounds, on the average, in a month by reducing his/her fat intake and by exercising consistently. Suppose weight loss has anormal distribution. Let $X$ = the amount of weight lost (in pounds) by a person in a month. Use a standard deviation of 2 pounds. $X$ ~ $\mathrm{N\left(5, 2\right)}$ . Fill in the blanks.

Suppose a person lost 10 pounds in a month. The z-score when $\mathrm{x = 10}$ pounds is $\mathrm{z = 2.5}$ (verify). This z-score tells you that $\mathrm{x = 10}$ is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).

This z-score tells you that $\mathrm{x = 10}$ is 2.5 standard deviations to the right of the mean 5 .

Suppose a person gained 3 pounds (a negative weight loss). Then $z$ = __________. This z-score tells you that $\mathrm{x = -3}$ is ________ standard deviations to the __________ (right or left) of the mean.

$z$ = -4 . This z-score tells you that $\mathrm{x = -3}$ is 4 standard deviations to the left of the mean.

Suppose the random variables $X$ and $Y$ have the following normal distributions: $X$ ~ $\mathrm{N\left(5, 6\right)}$ and $\mathrm{Y ~ N\left(2, 1\right)}$ . If $\mathrm{x = 17}$ , then . (This was previously shown.) If $\mathrm{y = 4}$ , what is $z$ ?

$z=\frac{y-\mu }{\sigma }=\frac{4-2}{1}=2\phantom{\rule{20pt}{0ex}}\text{where μ=2 and σ=1.}$

The z-score for $\mathrm{y = 4}$ is $\mathrm{z = 2}$ . This means that 4 is $\mathrm{z = 2}$ standard deviations to the right of the mean. Therefore, $\mathrm{x = 17}$ and $\mathrm{y = 4}$ are both 2 (of their ) standard deviations to the right of their respective means.

The z-score allows us to compare data that are scaled differently. To understand the concept, suppose $X$ ~ $\mathrm{N\left(5, 6\right)}$ represents weight gains for one group of people who are trying to gain weight in a 6 week period and $Y$ ~ $\mathrm{N\left(2, 1\right)}$ measures the same weight gain for a second group of people. A negative weight gain would be a weight loss.Since $\mathrm{x = 17}$ and $\mathrm{y = 4}$ are each 2 standard deviations to the right of their means, they represent the same weight gain relative to their means .

## The empirical rule

If $X$ is a random variable and has a normal distribution with mean $µ$ and standard deviation $\sigma$ then the Empirical Rule says (See the figure below)
• About 68.27% of the $x$ values lie between -1 $\sigma$ and +1 $\sigma$ of the mean $µ$ (within 1 standard deviation of the mean).
• About 95.45% of the $x$ values lie between -2 $\sigma$ and +2 $\sigma$ of the mean $µ$ (within 2 standard deviations of the mean).
• About 99.73% of the $x$ values lie between -3 $\sigma$ and +3 $\sigma$ of the mean $µ$ (within 3 standard deviations of the mean). Notice that almost all the $x$ values lie within 3 standard deviations of the mean.
• The z-scores for +1 $\sigma$ and –1 $\sigma$ are +1 and -1, respectively.
• The z-scores for +2 $\sigma$ and –2 $\sigma$ are +2 and -2, respectively.
• The z-scores for +3 $\sigma$ and –3 $\sigma$ are +3 and -3 respectively.

The Empirical Rule is also known as the 68-95-99.7 Rule.

Suppose $X$ has a normal distribution with mean 50 and standard deviation 6.

• About 68.27% of the $x$ values lie between -1 $\sigma$ = (-1)(6) = -6 and 1 $\sigma$ = (1)(6) = 6 of the mean 50. The values 50 - 6 = 44 and 50 + 6 = 56 are within 1 standard deviation of the mean 50. The z-scores are -1 and +1 for 44 and 56, respectively.
• About 95.45% of the $x$ values lie between -2 $\sigma$ = (-2)(6) = -12 and 2 $\sigma$ = (2)(6) = 12 of the mean 50. The values 50 - 12 = 38 and 50 + 12 = 62 are within 2 standard deviations of the mean 50. The z-scores are -2 and 2 for 38 and 62, respectively.
• About 99.73% of the $x$ values lie between -3 $\sigma$ = (-3)(6) = -18 and 3 $\sigma$ = (3)(6) = 18 of the mean 50. The values 50 - 18 = 32 and 50 + 18 = 68 are within 3 standard deviations of the mean 50. The z-scores are -3 and +3 for 32 and 68, respectively.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!