Theory:
To a better understand of how the experiment is analyzed, let explain some
important concept of probability and statistic as well the concepts of Mass flow and
Volume flow. It’s important that the reader have already knowledge about the basic
concept of probability and statistic to have a better understanding of the theory presented
here.
To make an easy explanation, here are presented some concepts that are used in
the experiment:
Probability and statistics concepts

Discrete distribution

Continuous distribution

Normal distribution

Mean Value

Standard Deviation

Variance

variation Coefficient

Bias coefficient

Chouvernet criteria

Bias estimator
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Engineering concepts

Mass flow

Volume Flow
Probability: Discrete and continuous distributions
An individual coin toss or the roll of a die is a random event, if repeated many
times the sequence of random events will exhibit certain statistical patterns, which can be
studied and predicted. As a mathematical foundation for statistics, probability theory is
essential to many human activities that involve quantitative analysis of large sets of data.
Discrete Distribution
“There are two type of probability distribution: Discrete distributions and Continuous
distributions. “A discrete probability function is a function that can take a discrete
number of values (not necessarily finite). This is most often the non-negative integers or
some subset of the non-negative integers. There is no mathematical restriction that
discrete probability functions only be defined at integers, but in practice this is usually
what makes sense. For example, if you toss a coin 6 times, you can get 2 heads or 3 heads
but not 2 1/2 heads. Each of the discrete values has a certain probability of occurrence
that is between zero and one. That is, a discrete function that allows negative values or
values greater than one is not a probability function. The condition that the probabilities
sum to one means that at least one of the values has to occur”[1].
Continuous distribution
This distribution is the one that our experiment has as well the normal distribution
A continuous probability must satisfy the following properties:
1. The probability that x is between two points a and b is𝑝[𝑎 ≤
𝑏
𝑥 ≤ 𝑏 = ∫𝑎 𝑓(𝑥)𝑑𝑥.
2. It is non-negative for all real x.
∞
3. The integral of the probability function, that is ∫−∞ 𝑓(𝑥)𝑑𝑥 = 1
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“Since continuous probability functions are defined for an infinite number of
points over a continuous interval, the probability at a sigle point is always zero.
Probabilities are measured over intervals, not single points. That is, the area under the
curve between two distinct points defines the probability for that interval. This means that
the height of the probability function can in fact be greater than one. The property that the
integral must equal one is equivalent to the property for discrete distributions that the sum
of all the probabilities must equal one”. [1]
Normal Distribution
The normal distribution is one of the most important distributions in probability
and statistic. There are a lot of behaviors that have this distribution. Height, weight and
other animal and human characteristic, measures of errors in scientific experiments,
anthropometrics measures of fossil and other.
A continuous fortuitous variable x is told that have a normal distribution with a
expected value μ, and standard deviation σ, if the function of probabilistic density is
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f(x)= (𝜎 ∗ √2𝜋) 𝑒
(𝑥−𝜇)²
2𝜎²
.
The graphic of the function is symmetric respect with the mean value or expected value
and the standard deviation determine the height and width of the graph. [2]
Figure 1: Symmetrical Graphical Function
Normal Distribution Graphic
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Mean Value
The mean value μ indicated the value where are concentrate the mayor probabilities. Is
the same as the arithmetic mean.
𝑥̅ =
∑𝑛
𝑖=1 𝑥𝑖
𝑛
.
Standard Deviation
The standard deviation σ, is a measure of the dispersion of the data around the mean
value. Bigger values of σ give a dispersion of data (a flatten graphic, more errors). Small
values give concentrate values near the mean value.
𝜎 = √𝑠²
Variance
The square of the standard deviation is known as the variance. The coefficient of
variation is another parameter that measures the dispersion and is defined as the ratio of
standard deviation between the expected value.
𝑠² = 𝜎² =
∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅ )²
𝑛−1
Figure 2: Function with different variance
Variation coefficient
The coefficient of variation compares the spread between two distinct
populations, and even compares the variation product of two different variables (which
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may come from the same population).
These variables may have different units, for example, we can determine whether the data
collected to measure the volume of water of a hydraulic bench, vary more than the data
taken by measuring the temperature of the liquid contained in the bench. The volume in
cubic centimeters and will measure the temperature in degrees Celsius.
“The coefficient of variation eliminates the dimensionality of the variables and takes into
account the ratio between a measure of trend, or standard deviation”. [3]
𝜎
𝐶𝑣 = 𝜇
Bias Coefficient
Bias coefficient determines the degree of asymmetry (lengthening of the
distribution
to the left or right). To determine the bias of a frequency distribution is used:
𝛾
𝐶𝑠 = 𝑆3
Figure 3: Direction of asymmetry
Bias Estimator
“Let θ* be an estimator of a parameter θ of a probability distribution. θ* is said to
be unbiased if its expectation is equal to the value of the parameter :
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E[θ*] = θ
In other words, the value of an unbiased estimator is, on the average, equal to the value to
be estimated.
If the expectation of the estimator is different from the value of the parameter, the
estimator is said to be "biased", and its bias is then defined as the difference between the
expectation of the estimator and the true value of the parameter:
Bias = E[θ*] - θ
The value of the bias depends, in general, on the value of the parameter”. [4]
𝑛
𝑛
𝛾=
∑(𝑥𝑖 − 𝑥̅ )³
(𝑛 − 1)(𝑛 − 2)
𝑖=1
Chauvenet criterion
Its specifies that a reading can be eliminated if a deviation from the average value
divided by the standard deviation is greater than the tabulated values in the data.
𝑋̇
Dmax = 𝜎
Engineering Concept
In this experiment we work with two concepts that are the one who are measured in the
experiment.
The first one is the Volume Flow rate is a measure of the volume of fluid passing a point
in the system per unit time.
𝑣
𝑣̇ = 𝑠
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The second one is the Mass Flow rate. The mass flow rate is the amount of mass that
pass through a pipe or, in this case, the mass that is in the Hydraulic Bench per unit of
time.
𝑚̇ = 𝜌𝑣̇
The density is the amount of material mass per unit volume.
𝜌=
𝑚
𝑣
Now with this concepts clear will be easier to understand this experiment.
Assumption:
Our first assumption is that the volume flow rate will not be constant, because the
experiment data is collected by student observation. So the values obtain will differ one
from another. The temperature will not have any significant change from the beginning of
the experiment to the end of the experiment, because the experiment is done in a
controlled room where the temperature is constant. According to the theory of statistic
analysis the expected value within the measured data of time of mass flow rate, should be
the average measure that also should be the same for the measure with the most occurring
probability.
Experimental Method
The procedure were obtain from manual of laboratory [5]
1. Each group will take 10 measurements of the time to fill the weight tank with a known weight
of water.
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2. Take the temperature of the water at the beginning, middle and end of the measurements.
3. Find in a table the density of water corresponding to the mean value of temperature during the
experiment.
4. Compute the volumetric flow for each time measurement.
5. Compute the arithmetic mean value, amplitude, variance, and standard deviation.
6. Use the Chauvenet criteria for eliminate the incorrect data.
7. Plot the histogram with the best class interval.
8. Use the empirical formulas to corroborate that the data correspond to a normal distribution
behavior.
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