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Introduction to Chemical Analysis: Errors, Significant Figures, and Descriptive Statistics, Study notes of Science education

A comprehensive introduction to chemical analysis, covering key concepts such as errors, significant figures, and descriptive statistics. It delves into the types of errors encountered in experimental data, including random, systematic, and gross errors. The document also explains the rules for significant figures in calculations and explores various measures of descriptive statistics, such as measures of frequency, dispersion, position, and inferential statistics. It is a valuable resource for students and researchers seeking a foundational understanding of chemical analysis principles.

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Definitions of Basic Terms in Analytical
Chemistry
Interference: An effect which alters or obscures the
behavior of an analyte in an analytical procedure (Sources
contaminants, reagents, instrumentation used for the
measurements)
Standards: Materials containing a known concentration
of a substance. They provide a reference to determine
unknown concentrations or to calibrate analytical
instruments.
Primary Standard: A substance whose purity and
stability are particularly well-established and with which
other standards may be compared. It is a pure substance
which reacts in a quantitative and known stoichiometric
manner with the analyte or a reagent.
Primary standard is a reagent that is extremely pure,
stable, has no waters of hydration, and has a high
molecular weight.
Criteria:
1. High Purity (testable easily)
2. High Stability in condition it is stored
(nonhygroscopic, non-volatile, not deliquescent,
unreactive towards atmosphere)
3. Larger molecular/equivalent weight (to decrease
relative error in weighing)
4. Cost-effectiveness
Certified reference material (CRM): A material that is
verified to contain a known amount of analyte(s) or to
have known physical properties. Usually available from
commercial suppliers. Also referred to simply as reference
material (RM) 4 See http://nist.gov/srm/definitions.cfm
for more details.
Standardization: Determination of the concentration of
an analyte or reagent solution from its reaction with a
standard or primary standard.
Procedure: A description of the practical steps involved in
an analysis.
Method: The overall description of the instructions for a
particular analysis.
Technique: The principle upon which a group of methods
is based (AAS, XPS etc.)
Speciation analysis: The determination of the specific
forms of an analyte. Common examples are elemental
mercury versus organomercury and different oxidation
states such as Cr(III) versus Cr(VI).
Method development. Determining the experimental
conditions for sample collection, preparation, and
measurement that produce accurate and repeatable
results.
Validation of Method:
- to validate the method by analyzing standards
which have an accepted analyte content, and a matrix
similar to that of the sample.
- Performing control experiments to verify the
accuracy, sensitivity, specificity, and reproducibility of test
methods.
Signal: The detector output that is displayed or recorded.
Noise: Random fluctuations in the signal. Usually
quantified using the standard deviation of multiple
measurements of a blank.
Selectivity: The ability of a method or instrument to
measure an analyte in the presence of other constituents of
the sample or test portion. (Pure & Appl. Chem. 2001, 73(8),
1381-1386.]
Sensitivity: The change in the response from an analyte
relative to a small variation in the amount being
determined. The sensitivity is equal to the slope of the
calibration curve (the change in detector signal versus the
change in amount of analyte), being constant if the curve is
linear. In other words, it is the ability of a method to
facilitate the detection or determination of an analyte. A
higher sensitivity may allow measurement of a lower
analyte concentration, depending on the signal to-noise
ratio.
Determination: A quantitative measure of an analyte
with an accuracy of considerably better than 10% of the
amount present.
Estimation: A semi-quantitative measure of the amount
of an analyte present in a sample, with accuracy no better
about 10% of the amount present.
Detection Limit: The smallest amount or concentration of
an analyte that can be detected by a given procedure and
with a given degree of confidence. It is the minimum
measured concentration at which an analyte may be
reported as being detected in the test portion or sample.
Limit of linearity (LOL): The concentration at which the
signal deviates from linearity.
Limit of quantitation (LQ): An analytical value, X above
which quantitative determinations are possible with a
given minimum precision.
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Definitions of Basic Terms in Analytical

Chemistry

Interference: An effect which alters or obscures the behavior of an analyte in an analytical procedure (Sources contaminants, reagents, instrumentation used for the measurements) Standards: Materials containing a known concentration of a substance. They provide a reference to determine unknown concentrations or to calibrate analytical instruments. Primary Standard: A substance whose purity and stability are particularly well-established and with which other standards may be compared. It is a pure substance which reacts in a quantitative and known stoichiometric manner with the analyte or a reagent. Primary standard is a reagent that is extremely pure, stable, has no waters of hydration, and has a high molecular weight. Criteria:

  1. High Purity (testable easily)
  2. High Stability in condition it is stored (nonhygroscopic, non-volatile, not deliquescent, unreactive towards atmosphere)
  3. Larger molecular/equivalent weight (to decrease relative error in weighing)
  4. Cost-effectiveness Certified reference material (CRM): A material that is verified to contain a known amount of analyte(s) or to have known physical properties. Usually available from commercial suppliers. Also referred to simply as reference material (RM) 4 See http://nist.gov/srm/definitions.cfm for more details. Standardization: Determination of the concentration of an analyte or reagent solution from its reaction with a standard or primary standard. Procedure: A description of the practical steps involved in an analysis. Method: The overall description of the instructions for a particular analysis. Technique: The principle upon which a group of methods is based (AAS, XPS etc.) Speciation analysis: The determination of the specific forms of an analyte. Common examples are elemental mercury versus organomercury and different oxidation states such as Cr(III) versus Cr(VI). Method development. Determining the experimental conditions for sample collection, preparation, and measurement that produce accurate and repeatable results. Validation of Method: - to validate the method by analyzing standards which have an accepted analyte content, and a matrix similar to that of the sample. - Performing control experiments to verify the accuracy, sensitivity, specificity, and reproducibility of test methods. Signal : The detector output that is displayed or recorded. Noise: Random fluctuations in the signal. Usually quantified using the standard deviation of multiple measurements of a blank. Selectivity: The ability of a method or instrument to measure an analyte in the presence of other constituents of the sample or test portion. (Pure & Appl. Chem. 2001, 73(8), 1381-1386.] Sensitivity: The change in the response from an analyte relative to a small variation in the amount being determined. The sensitivity is equal to the slope of the calibration curve (the change in detector signal versus the change in amount of analyte), being constant if the curve is linear. In other words, it is the ability of a method to facilitate the detection or determination of an analyte. A higher sensitivity may allow measurement of a lower analyte concentration, depending on the signal to-noise ratio. Determination: A quantitative measure of an analyte with an accuracy of considerably better than 10% of the amount present. Estimation: A semi-quantitative measure of the amount of an analyte present in a sample, with accuracy no better about 10% of the amount present. Detection Limit : The smallest amount or concentration of an analyte that can be detected by a given procedure and with a given degree of confidence. It is the minimum measured concentration at which an analyte may be reported as being detected in the test portion or sample. Limit of linearity (LOL): The concentration at which the signal deviates from linearity. Limit of quantitation (LQ) : An analytical value, X ₁above which quantitative determinations are possible with a given minimum precision.

Equivalent: That amount of a substance which, in a specified chemical reaction, produces, reacts with, or can be indirectly equated with one mole (6.023 x 1022) of hydrogen ions. This confusing term is obsolete, but its use is still to be found in some analytical laboratories. Selecting and Handling Reagents and other Chemical Purity of reagents - The purity of reagents influences the accuracy of analysis. Reagent grade : conform to the minimum standards set forth by the Reagent Chemical Committee of the American Chemical Society (ACS). EXAMPLE: Tollens reagent The reagent consists of a solution of silver nitrate, ammonia and some sodium hydroxide (to maintain a basic pH of the reagent solution). Primary-standard carefully analyzed by the supplier. The National Institute of Standards and Technology (NIST) is an excellent source. EXAMPLE: Standardization of silver nitrate. Sodium chloride is used as the primary standard for this purpose. Some Examples: Sodium carbonate (Na2CO3) Potassium hydrogen phthalate (KHP): C8H5KO4, mol wt.= 204.23 g/mol, Potassium hydrogen iodate: KH(IO3)2, mol wt. = 389. g/mol Sodium tetraborate Na2B4O7, Special-purpose reagent chemicals are prepared for a specific application such as solvents spectrophotometry and for high- performance liquid chromatography. Rules for Handling Reagents and Solutions 1: Select the best grade of chemical available. Pick the smallest bottle that is sufficient to do the job. 2: Replace the top of every container immediately after removing reagent 3: Hold the stoppers of reagent bottles between your fingers. Never set a stopper on a desktop 4: Unless specifically directed otherwise, never return any excess reagent to a bottle. 5: Never insert spatulas, spoons, or knives into a bottle that contains a solid chemical. Instead, shake the capped bottle vigorously or tap it gently against a wooden table to break up an encrustation. Then pour out the desired quantity. 6: Keep the reagent shelf and the laboratory balance clean and neat. Clean up any spills immediately. 7: Follow local regulations concerning the disposal of surplus reagents and solutions. Cleaning and marking of laboratory ware 1: Each vessel that holds a sample must be marked. Special marking Inks are available for porcelain surfaces. A saturated solution of iron (III) chloride can also be used for marking. 2: Every apparatus must be thoroughly washed with a hot detergent solution and then rinsed, initially with large amounts of tap water and finally with several small portions of deionized water 3: Properly cleaned glassware will be coated with a uniform and unbroken film of water. Do not dry the interior surfaces of glassware. 4: An organic solvent, such as methyl ethyl ketone or acetone, may be effective in removing grease films. Evaporating Liquids

Beaker – cylindrical container with a flat bottom.

Vessels in which liquid is placed so it can be stirred,

mixed or heated.

Alcohol Lamp - used for heating, sterilization, and

combustion in a laboratory.

Evaporating Dish – used as reaction vessels, or

for the separation of the solute from a solution

through crystallization.

Measuring mass

Analytical Balance - highly sensitive lab instruments

designed to accurately measure mass. An analytical

balance has a maximum capacity that ranges from 1g

to several kilograms and a precision at maximum

capacity of a least 1 part in 105.

Importance of Identifying Chemicals and

Laboratory apparatuses used in an analysis

The paramount to use the right kind of laboratory

apparatus for each and every experiment. Whether to

use an electronic pipette or a manual one for a

certain experiment, which type of scales to use for

the experiment, are questions that you should have

answers to before starting off with the experiment.

There are four different ways chemicals can

enter the body. These are:

Inhalation: Chemicals that take form in gas, vapour

or particulates are easily inhaled. These chemicals

can absorb into the respiratory tract and can head

into the bloodstream and organs. This is often noted

as the most common way the body absorbs harmful

chemicals.

Injection : This doesn't necessarily mean directly

injecting chemicals into your bloodstream, but if you

have a cut or other tear in the skin, chemicals can be

absorbed this way.

Skin/Eye absorption : Chemical contact with skin

can result in mild dermatitis, or a rash. However,

chemicals can also be absorbed into the bloodstream

this way. Eyes are also sensitive to most chemicals, so

safety glasses must be worn when conducting work

with chemicals. Another common scenario that

causes eye contact to chemicals (especially if not

wearing appropriate safety glasses) is wiping or

rubbing at your eyes during chemical exposure.

Ingestion: Like with inhalation or skin/eye

absorption, ingestion can cause the toxic chemicals

to travel to the organs. When. conducting work in

areas where ingestion is likely, like confined spaces,

it's important to have an entry & exit plan, and the

proper PPE for the job.

Accidents with hazardous chemicals can happen

quickly and may be quite severe. The key to

prevention of these accidents is awareness.

ERRORS IN CHEMICAL ANALYSIS

Error – defined as the difference between the true result (or accepted true result) and the measured result. Precise

  • Closeness of replicate measurements with each other. Accurate - The closeness of a given measurement to the true value. Significant figure - are the digits of a number that are meaningful in terms of accuracy or precision. Random error - Caused by uncontrollable or unknown fluctuations in variables that may affect experimental results. Systematic error – errors that are known and controllable errors or errors that affects the accuracy of data sets. Gross error - Cause large errors leading to outlier data. Absolute error - Provide an idea whether the measured value is higher or lower compared to the true value. Relative error - Simply the absolute error divided by the true value and can be expressed in percent.

Significant Figures are the digits of a number that are meaningful in terms of accuracy or precision. The importance of significant figures in reporting measurement is the number of significant figures corresponds to all digits that are certain in a measurement plus one uncertain digit. VALUES SF 1.) 45 TWO 2.) 0. TWO 3.) 7.4220 FIVE 4.) 5002 FOUR 5.) 3800 FOUR Determining the number of significant figures RULES FOR SIGNIFICANT FIGURES

  1. All non-zero digits are significant. Example: 198745 – 6 significant figures
  2. All zeros that occur between any two non-zero digits are significant. Example: 108.0097 – 7 significant figures
  3. All zeros that are on the right of a decimal point and to the left of a non-zero digit is never significant. Example: 0.00798 -3 significant figures 4.All zeros that are on the right of a decimal point are significant, only if, a non-zero digit does not follow them. Example: 30.00 – 4 significant figures
  4. All the zeros that are on the right of the last non- zero digit, after the decimal point, are significant. Example: 0.0079800 - 5 significant figures
  5. All the zeros that are on the right of the last non- zero digit are significant if they come from a measurement. Example: 1090 - 4 significant figures SIGNIFICANT FIGURES IN CALCULATIONS Addition and Subtraction When adding or subtracting digits, the results shall have the same number of decimal places as the given measurement with the least number of decimal places. Otherwise, the result will have to be rounded off. Example: a. 4.2 + 8.236 + 7.91 = 20.346; 4.2 (1 decimal place (least)) Final Answer = 20. b. 15.6 – 7.27 = 8.33; 15.6 (1 decimal place (least)) Final Answer= 8. Multiplication and Division When multiplying and dividing numbers, the answer should bear the same number of significant figures, as the given measurement with the least number of significant figures. Otherwise, the result must be rounded off. EXAMPLE: (253 x 3.45)/72 = 12.1229167; 72 (least significant figure) Final Answer≈ 12 Logarithms and Antilogarithms
    1. The number of significant figures in the original number will be the number of decimal places of the logarithm of that number. EXAMPLE: log 3.425 x 10^-5 = -4. 4653
    2. The number of decimal places in the original number will be the number of significant figures to be reported in the antilogarithm of that number. EXAMPLE: log-1 22. 54 = 3.5 x 1022 Chemical calculations rounding off Rounding off should be done in final step only A number is rounded off to the required number of significant digits by leaving one or more digits from the right. When the first digit in left is less than 5, the last digit held should remain constant. When the first digit is greater than 5, the last digit is rounded up. When the digit left is exactly 5, the number held is rounded up or down to receive an even number. When more than one digit is left, rounding off should be done as a whole instead of one digit at a time. Example: a. 46.5 + 23.25 = 69. ; 46.5 (least significant decimal) Final Answer = 69. b. 46.514 + 23.15 = 69. ; 23.15 (least significant decimal) Final Answer = 69. ACCURACY AND PRECISION Accuracy is the closeness of a given measurement to the true value. Example:

⚫ natural variations in real world or experimental contexts. ⚫ imprecise or unreliable measurement instruments. ⚫ individual differences between participants or units Systematic error (Determinate) are those errors that are known and controllable errors. Error affects the accuracy of data sets. TYPES OF SYSTEMATIC ERROR OFFSET ERRORS occurs when a scale isn’t calibrated to a correct zero point. It’s also called an additive error or a zero-setting error. EXAMPLE: When measuring participants’ wrist circumferences, you misread the “2” on the measuring tape as a zero-point. All of your measurements have an extra 2 centimeters added to them. SCALE FACTOR ERROR is when measurements consistently differ from the true value proportionally (e.g. by 10%). It’s also referred to as a correlational systematic error or a multiplier error. EXAMPLE: A weighing scale consistently adds 10% to each weight. A true weight of 10 kg is recorded as 11 kg, while a true weight of 40 kg is recorded as 44 kg CLASSIFICATION OF SYSTEMATIC ERROR Instrumental error are errors caused by poor instrument condition and calibration. Method error are caused by poor outcomes caused by substandard conditions of chemicals and reactions. Example: In volumetric analysis, the use of improper indicator leads to wrong results. Personal error are caused by personal limitations of the analyst such as failing to follow procedures properly, among others. Example: Parallax Error The error/displacement caused in in the apparent position of the object due to the viewing angle that is other than the angle that is perpendicular to the object. DETECTION OF SYSTEMATIC ERROR Systematic errors can be detected by: Analysis of reference standards reference standards are chemicals of known concentration and purity Example: pH buffer solution Analysis of blank samples blank samples contain all reagents used in the analysis other than the sample. Third Party Analysis allows other chemists to analyze the sample. Variation of sample size to check for constant or proportional errors. ⚫ Constant error - an average of the errors over the range of all data. Errors that do not change in magnitude as sample size increases. ⚫ Proportional Error - an error that is dependent on the amount of change in a specific variable. Errors that increase in magnitude as sample size increase. Example: Titration GROSS ERRORS causes large errors leading to outlier data. These are errors that are so serious (i.e. large in magnitude) that they cannot be attributed to either systematic or random errors associated with the sample instrument, or procedure. Example: When the contents of a mixture is spilled when it is being boiled. The loss brought about by spilling causes an error in the amount of the analyte present in the sample being boiled. 02: A. INTRODUCTION TO STATISTICS WHAT IS STATISTICS? A branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. The science that deals with the collection, classification, analysis, and interpretation of numerical facts or data, and that, by use of mathematical theories of probability, imposes order and regularity on aggregates of more or less disparate elements. WHAT IS VARIABLE?  a variable is a characteristics or condition that can change or take on different values.  most research begins with a general question about the relationship between two variables for a specific group of individuals. 03: TYPES OF VARIABLES DISCRETE VARIABLE - a variable that may assume only a countable, and usually finite, number of values (such as

class size). It consists of indivisible categories. No values can exist between two neighboring categories. Example: Number of planets around the Sun or Number of students in a class CONTINUOUS VARIABLE - Continuous variables would take forever to count (such as time or weight) are infinitely divisible into whatever units a researcher may choose. Example: time can be measured to the nearest minute, second, half-second or Age (you can’t count “age”, you could be: 25 years, 10 months, 2 days, 5 hours, 4 seconds, 4 milliseconds, 8 nanoseconds, 99 picosends…and so on) REAL LIMIT - To define the units for a continuous variable, a researcher must use real limits which are boundaries located exactly half-way between adjacent categories. Example: a test score of 95 has the lower real limit of 94. and the upper real limit of 95.4 since any value within that range will equal 95 when rounded to a whole number. MEASURING VARIABLE - To establish relationships between variables, researchers must observe the variable and recorded their observations. This requires that the variables be measured.

 The process of measuring a variable requires a set of

categories called a scale of measurement and a process that classifies each individual into one category. 04: TYPES OF MEASURING VARIABLE NOMINAL SCALE - is an unordered set of categories identified only by name. Nominal measurements only permit you to determine whether two individuals are the same or different. Example: blood type ORDINAL SCALE - is an ordered set of categories. Ordinal Measurement tell you the direction of difference between two individuals. Example: satisfaction rating (“extremely dislike”, “dislike”, “neutral”, “like”, “extremely like”) INTERVAL SCALE - is an ordered series of equal sized categories. Interval measurement identify the direction and magnitude of a difference. The zero point is located arbitrarily on an interval scale. Variables that have familiar, constant, and computable differences are classified using the Interval scale. Example: in the temperature, there is no point where the temperature can be zero. Zero degrees F does not mean the complete absence of temperature. INTERVAL SCALE - is an interval scale where a value of zero indicates none of the variable. Ratio measurements identify the direction and magnitude of difference and allow ratio comparison of measurements. Example: the temperature outside is 0-degree Celsius. 0 degree doesn’t mean it’s not hot or cold, it is a value. 05: DATA TYPES NOMINAL (CATEGORICAL) - No comparison is defined. Example: Gender (male & female) ORDINAL - Comparable but the difference is not defined Example: socio economic status (“low income”, “middle income”,” high income”) INTERVAL - Deduction and addition is define but not division Example: calculate intelligence score in an IQ test. RATIO - a form of numerical data which is quantitative in nature. Example: HEIGHT (What is your height in feet and inches?) -Less than 5 feet. -5 feet 1 inch – 5 feet 5 inches -5 feet 6 inches- 6 feet -More than 6 feet EXPERIMENTS

  • The goal of an experiment is to demonstrate a cause-and-effect relationship between two variables.
  • In an experiment, one variable is manipulated to create treatment conditions and the second variable is observed and measured to obtain scores for a group of individuals in each of the treatment conditions.
  • The manipulated variable is called the independent variable and the observed variable is the dependent variable 06: POPULATION VS. SAMPLE POPULATION - the entire group of individuals Example: a researcher may be interested in the relation between class size (variable 1) and academic performance (variable
  1. for the population of third grade children. SAMPLE - a small part or quantity intended to show what the whole is like  Usually, populations are so large that a researcher cannot examine the entire group. Therefore, a sample is selected to represent the population in a research study. The goal is to use the results obtained from the sample to help answer questions about the population. EXAMPLE: from the third-grade children the researcher will choose 5 children.

VARIANCE

EXAMPLE: Data set: 46, 69, 32, 60, 52, 41 Step 1: Find the mean  x̅ = ∑ x / n = (46 + 69 + 32 + 60 + 52 + 41) ÷ 6 = Step 2: Find each score’s deviation from the mean. Score Deviation from the mean 46 46 – 50 = - 69 69 – 50 = 19 32 32 – 50 = - 60 60 – 50 = 10 52 52 – 50 = 2 41 41 – 50 = - VARIANCE EXAMPLE: Data set: 46 69 32 60 52 41 Step 5: Divide the sum of squares by n – 1 or N Variance σ = 886 ÷ (6 – 1) = 886 ÷ 5 = 177 CONFIDENCE INTERVAL  is a specific interval estimate of a parameter determined by using data obtained from a sample.  The confidence interval is the preferred method for describing the range of uncertainty in a value. The confidence interval is expressed as a range of uncertainties at a stated percent confidence. This percent confidence reflects the percent certainty that the value is within the stated range.

CONFIDENCE INTERVAL

WHEN DO YOU USE CONFIDENCE INTERVALS?

You can calculate confidence intervals for many kinds of statistical estimates, including: Proportions Population means Differences between population means or proportions Estimates of variation among groups

4. MEASURES OF POSITION  Use this when you need to compare scores to a normalized score  Describe how scores fall in relation to one another. Relies on standardized scores  Use percentile ranks, quartile rank to measure position PERCENTILE RANKS - the percentage of scores in its frequency distribution that are equal to or lower than it.

EXAMPLE: a test score that is greater than 75% of the scores of people taking the test is said to be at the 75th percentile, where 75 is the percentile rank. QUARTILE RANKS - a type of quantile which divides the number of data points into four parts, or quarters, of more- or-less equal size. FORMULA FOR QUARTILE: Lower Quartile (Q1) = (N+1) * 1 / 4 Middle Quartile (Q2) = (N+1) * 2 / 4 Upper Quartile (Q3)= (N+1) * 3 / 4 INFERENTIAL STATISTICS - Are methods for using sample data to make general conclusions (inferences) about populations Inferential statistics have two main uses:  making estimates about populations. Example: the mean SAT score of all 11th graders  testing hypotheses to draw conclusions about populations. Example: the relationship between SAT scores and family income 09: TYPES OF INFERENTIAL STATISTICS

  1. T-TEST  is a statistical test that is used to compare the means of two groups.  It is often used in hypothesis testing to determine whether a process or treatment actually has an effect on the population of interest, or whether two groups are different from one another.  The independent variable is Qualitative, and the dependent variable is Quantitative  Is used when the data follows a student t distribution and the sample size is lesser than 30. It is used to compare the sample and population mean when the population variance is unknown. What type of t-test should I use? When choosing a t-test, you will need to consider two things: whether the groups being compared come from a single population or two different populations, and whether you want to test the difference in a specific direction TWO SAMPLES COMPARED A larger t-value shows that the difference between group means is greater than the pooled standard error, indicating a more significant difference between the groups.  If the groups come from a single population (e.g. measuring before and after an experimental treatment), perform a paired t-test.  If the groups come from two different populations (e.g. two different species, or people from two separate cities), perform a two-sample t-test (a.k.a. independent t-test).  If there is one group being compared against a standard value (e.g. comparing the acidity of a liquid to a neutral pH of 7), perform a one-sample t-test. 2. Q-TEST - a nonparametric inferential test that enables a researcher to assess the significance of the differences among two or more matched samples on a dichotomous outcome. - It is a way to find outliers in very small, normally distributed, data sets. Where: x1 is the smallest (suspect) value, x2 is the second smallest value, and xn is the largest value.