Soil Science Fundamentals

Cheatsheet Content

### Pioneers of Soil Science - **Justus von Liebig:** - Father of the fertilizer industry - Early concepts on soil - **Friedrich Albert Fallou:** - Systematic approach for soil science - Recognition as a discipline - **Vasily Dokuchaev:** - Father of soil science - Soil formation and classification ### Hydrologic Cycle and Soil Water enters the soil, interacts with soil particles, and leaves the soil (through evaporation, transpiration, or leaching). ### Soil: Definition - Biologically active, porous medium that has developed in the uppermost layer of Earth's crust. - The continental crust is primarily composed of igneous, metamorphic, and sedimentary rocks, referred to as "sial" (silica and aluminum). #### Soil Properties influenced by the hydrologic cycle: - Moisture - Structure - Texture - Bulk Density - Temperature - Color - pH - Horizon/Depths ### Importance of Soil - **Life-Supporting Layer:** Forms a very thin interface between the crust and atmosphere. - **Resources for Plants and Animals:** Provides oxygen, temperature, water, carbon, and nutrients. - **Non-agricultural Uses:** - Recreation: Playgrounds, golf courses, parks - Engineering projects: Construction (buildings, highways, rail, roads, bridges) - Waste disposal ### The Ideal Soil - A natural body consisting of layers (soil horizons) of mineral constituents of variable thickness, differing from parent materials in morphological, physical, chemical, and mineralogical characteristics. - Composed of altered broken rock particles due to weathering and erosion. - Differs from parent rock due to interactions between the lithosphere, atmosphere, hydrosphere, and biosphere. #### Major components of soil: 1. Eroded rock 2. Mineral nutrients 3. Decaying organic matter 4. Water (25%) 5. Air (25%) 6. Living organisms (10%) ### Soil Formation Factors - **Weathering:** Surface phenomenon (deterioration of rocks/minerals). - **Erosion:** Bulk phenomenon. - **Factors:** Parent Material, Vegetation, Climate, Topography, Time. #### Physical properties (what we can see and feel): - Soil color - Sand and aggregates - Water content - Soil texture - Compactness of soil ### Terms Associated with Soil Creation - **Infiltration:** Downward movement of water through the soil. - **Leaching:** Dissolving of minerals and organic matter in upper layers and carrying them to lower layers. - Soil type determines the degree of leaching and infiltration. ### Soil Characterization Protocols #### Basic: - Field Characterization - Bulk Density - Porosity - pH - Soil Moisture - Soil Temperature - Soil color #### Advanced: - Particle Size Distribution - Particle Density - Soil Fertility #### Optional: - Soil Infiltration ### Particle Size #### USDA (United States Department of Agriculture) Classification: | Soil fraction | Diameter | Description | |:--------------|:--------------------|:------------------| | Gravel | >2 mm | Coarse | | Sand | 0.05 - 2 mm | Gritty | | Silt | 0.002 - 0.05 mm | Floury | | Clay | ### Surface Area - Surface area increases while total volume remains constant. - Many chemical and physical processes take place on the particle surface. | Kind of Particle | Diameter of Particle | No. of Particles in 1 gram | Surface area of 1 gram | |:-----------------|:---------------------|:---------------------------|:-----------------------| | Sand | 2 mm | 90 | 11 cm² | | Silt | 0.002 to 0.05 mm | 9x10⁷ | 1130 cm² | | Clay | ### Soil Texture - Percentage of sand, silt, and clay a soil contains. - **Loam:** Roughly equal proportions of sand, silt, and clay. ### Soil Horizon - Horizon formation is a function of geological, chemical, and biological processes over long time periods. - Horizons are defined by physical features, color, and texture. - **O) Organic matter:** Litter layer of plant residues in relatively undecomposed form. - **A) Surface soil:** Layer of mineral soil with most organic matter accumulation and soil life. - **E) A horizon:** Eluviates (is depleted of) iron, clay, aluminum, organic compounds. - **B) Subsoil:** Accumulates iron, clay, aluminum, organic compounds. - **C) Parent rock:** Layer of big unbroken rocks. - **Typical Depths:** O: 0-2", A: 2-10", E: 10-20", B: 20-30", C: 30-48" ### Soil Profiles on Hillslopes - Thickness and composition of soil horizons vary with position on a hillslope and with water drainage. - **Poorly drained profiles (upper slopes):** Underlying rock may be exposed by surface erosion; nutrient-rich soils (A horizon) may accumulate at the toe-slope. - **Well-drained profiles (under forest cover):** Leached layers (E horizon) may be relatively thick, and surface erosion minimal. ### Soil Nomenclature #### Soil Orders: - Alfisols - Andisols - Aridisols - Entisols - Gelisols - Histosols - Inceptisols - Mollisols - Oxicols - Spodosols - Ultisols - Vertisols - Rocky Land - Shifting Sand - Ice/Glacier ### ICAR Classification (India) #### Soil Types in India: - Alluvial Soil - Black Soil (Regur Soil) - Red Soil - Laterite Soil - Arid and Desert Soil - Saline and Alkaline Soil - Peaty and Marshy Soil - Forest and Mountain Soils ### Bulk Density and Porosity - **Bulk Density:** Property of powders, granules, and soils. - Formula: $$\frac{\text{mass of oven dry soil}}{\text{volume of soil}}$$ - Depends on mineral makeup and degree of compaction. - **Porosity:** Measure of total pore space in soil. - Formula: $$\frac{\text{volume of voids}}{\text{solid + pore space}}$$ - Fraction between 0 and 1, or expressed as %. - Inversely related to bulk density: more pore space = lower bulk density. - As aggregation and clay content increase, bulk density decreases. - Clayey soils tend to have lower bulk densities and higher porosities than sandy soils. - **Particle Density:** Mass per unit volume of the soil solids only. ### Soil Organic Matter - Derived from plants and animals. - When litter and wood decay to the point where it's no longer recognizable, it's called **soil organic matter**. - When organic matter breaks down into stable humic substances that resist further decomposition, it's called **humus**. - Humus increases the soil's **cation exchange capacity**, storing nutrients by chelation, similar to clay particles. - Nutrient cations are accessible to plants and held in the soil, safe from leaching by rain or irrigation. ### Soil Color - Surface soil horizon color depends mainly on **organic matter content**. - Darker soil = more organic matter. - Organic matter imparts favorable properties: better aggregation, high water-holding capacity. - **Munsell Color book** used by soil scientists: - **Hue:** Basic/dominant color. - **Value:** Lightness or darkness. - **Chroma:** Color intensity. - **Well-drained soils:** Brown or red due to oxidation of Fe. - **Poorly drained soils (wetlands):** Black, gray, or even blue due to prolonged water saturation. ### Soil Water Capacity - **Saturation:** All pore spaces filled; function of soil properties. - **Field Capacity:** Macropores drained; function of soil properties. - **Wilting Point:** Plant cannot extract any more water from soil; function of plant and soil properties. ### Soil Processes - **Key:** - Orange: Storages - Green: Inputs - Blue: Outputs - Purple: Transfers - Red: Transformations ### Essential Nutrients #### Essential Elements Used in Relatively Large Amounts: - **Mostly from Air and Water:** Carbon (C), Hydrogen (H), Oxygen (O) - **From Soil Solids (NPK):** Nitrogen (N), Phosphorus (P), Potassium (K), Calcium (Ca), Magnesium (Mg), Sulfur (S) #### Essential Elements Used in Relatively Small Amounts: - **From Soil Solids:** Iron (Fe), Copper (Cu), Manganese (Mn), Zinc (Zn), Boron (B), Chlorine (Cl), Molybdenum (Mo) ### Macronutrients vs. Micronutrients #### Macronutrients: - **Mostly from air and water:** - Carbon (CO₂) - Hydrogen (H₂O) - Oxygen (O₂, H₂O) - **Mostly from soil solids:** - Nitrogen (NO₃⁻, NH₄⁺) - Phosphorus (H₂PO₄⁻, HPO₄²⁻) - Potassium (K⁺) - Calcium (Ca²⁺) - Magnesium (Mg²⁺) - Sulfur (SO₄²⁻) #### Micronutrients: - **Mostly from soil solids:** - Iron (Fe²⁺) - Manganese (Mn²⁺) - Boron (H₃BO₃) - Zinc (Zn²⁺) - Copper (Cu²⁺) - Chlorine (Cl⁻) - Cobalt (Co²⁺) - Molybdenum (MoO₄²⁻) - Nickel (Ni²⁺) ### Ions Present in Soil Solution - **Soil Solution:** Aqueous liquid found within a soil. | Elements | Symbol | Form Used by Plants | |:----------|:-------|:-------------------------------| | Sulfur | S | SO₃²⁻, SO₄²⁻ | | Carbon | C | CO₃²⁻, HCO₃⁻, CO₂ | | Hydrogen | H | H₂O | | Oxygen | O | O₂ | | Nitrogen | N | NH₄⁺, NO₂⁻, NO₃⁻ | | Phosphorus| P | HPO₄²⁻, H₂PO₄⁻ | | Potassium | K | K⁺ | | Calcium | Ca | Ca²⁺ | | Magnesium | Mg | Mg²⁺ | ### Solid-Liquid Interactions - Ions or molecules are: - Absorbed by roots - Leached away - Ions must first be released from solids by: 1. Cation exchange 2. Anion exchange 3. Ligand exchange 4. Dissolution / precipitation 5. Surface chelation ### Cation Exchange in Soils - **Soil Fertility:** - Replenishes the soil solution (feeds roots). - Holds nutrients against leaching. - Influences nutrient availability to roots. - **Environmental Impacts:** - De-activates toxins (metals, organics) - Filters percolating waters - **Cations Involved Include:** - **Nutrients:** NH₄⁺, K⁺, Ca²⁺, Mg²⁺, Fe²⁺, Zn²⁺, Cu²⁺ - **Pollutants:** Herbicides, insecticides, PCBs, toxic metals ### Types of Adsorption Complexes in Soil - **(1) Diffuse Ions:** Ions loosely held in solution near the surface. - **(2) Outer-Sphere Complex:** Ions hydrated and held electrostatically to charged sites on the surface. - **(3) Inner-Sphere Complex:** Ions directly bonded to the surface, often sharing electrons. ### Clay Mineral Structure (Montmorillonite) - The basic building block for the silica-dominated sheet is a unit composed of one silicon atom surrounded by four oxygen atoms. - Aluminum and/or magnesium ions are key cations in the second type of sheet. - The tetrahedral and octahedral sheets are the fundamental structural units of silicate clays, bound together within crystals by shared oxygen atoms into different layers. ### Charge Development on Clays - **Two main sources of charge:** Isomorphous substitution and pH-dependent charges. - **Isomorphous Substitution:** Main source of charge in silicate clays. - Substitution of one element for another in ionic crystals without changing the structure (e.g., Al³⁺ for Si⁴⁺ in tetrahedral coordination, Mg²⁺ or Fe²⁺ for Al³⁺ in octahedral coordination). - Charges developed are **permanent** and not pH-dependent. - **pH-Dependent Charges:** Main source of charge in kaolinite and metal oxides. - Charges are variable, either positive or negative, depending on soil pH. - In metal oxides, acid soils tend to develop positive charges due to protonation on oxide surfaces. ### Isomorphic Substitution in 2:1 Mineral Clay Crystal - **No Substitution:** Net charge = 0 - **With Isomorphic Substitution:** (e.g., Al³⁺ replaced by Mg²⁺) Net charge = -1 ### Charged Functional Groups on Model Humus - **Carboxyl group** - **Phenolic hydroxyl group** - **Alcoholic hydroxyl group** ### Soil Cation Exchange - **Cation Exchange:** The ability of soil to hold onto nutrients and prevent them from leaching beyond the roots. - Measured by the number of cation adsorption sites per unit weight of soil, or the sum total of exchangeable cations a soil can adsorb. - Higher cation exchange capacity = higher fertility level. ### Measuring the Cation Exchange Capacity of a Soil ### Principles of Cation Exchange #### Reversibility: $$ \text{Micelle} - \text{Na}^+ + \text{H}^+ \rightleftharpoons \text{Micelle} - \text{H}^+ + \text{Na}^+ $$ #### Charge Equivalence: $$ \text{Micelle} - \text{Ca}^{2+} + 2\text{H}^+ \rightleftharpoons \text{Micelle} - (\text{H}^+)_2 + \text{Ca}^{2+} $$ ### The Ratio Law - The ratio of ions on exchange sites is equal to the ratio of ions in the soil solution. ### Cation Exchange Influenced By: 1. **Strength of adsorption:** - $$\text{Al}^{3+} > \text{Ca}^{2+} > \text{Mg}^{2+} > \text{K}^+ = \text{NH}_4^+ > \text{Na}^+ > \text{H}^+$$ - Held tightly $\rightarrow$ easily replaced. - Based on valence charge and hydrated ionic radius. - Selectivity = $$\frac{\text{Charge of ion}}{\text{Size}}$$ 2. **Relative concentration of the cations in the Soil Solution.** ### Anion Exchange - Generally much smaller than Cation Exchange Capacity (CEC). - **Anions of interest in order of sorption strength:** - $$\text{H}_2\text{PO}_4^- > \text{MoO}_4^{2-} >> \text{SO}_4^{2-} >> \text{Cl}^- > \text{NO}_3^-$$ - Also organic anions (R-O⁻) and silicic acid H₂SiO₄⁻. - Kaolinitic clays may have AEC up to 5 cmol(-). - SOM in acid soils may provide AEC > 20 cmol(-). ### Units for Expressing CEC - **Traditional unit:** milli-equivalent (me/meq) per 100 grams: - me/100g - me = mmol/valence - **System International (SI) units:** - mmol(+) / kg soil - cmol(+) / kg soil = 10 mmol(+)/1000g = me/100g - **Exchangeable ions example:** - 4 me/100g Ca²⁺ = (4/2) mmol/100g Ca²⁺ = 2 mmol/100g Ca²⁺ - = 2 cmol/kg (Ca²⁺) = 4 cmol (+) /kg - CEC is expressed in milliequivalents (meq) per 100 g of **oven dry soil**. - **Equivalent weight** = $$\frac{\text{molecular wt (g)}}{\text{valence or charges as per formula}}$$ ### Other Processes of Soil-Water Interaction #### Ligand Exchange: - Property of amorphous and oxide minerals. - Example: $$\text{Al-OH} + \text{H}_2\text{PO}_4^- \rightarrow \text{Al-O-P-OH} + \text{OH}^-$$ - Not anion exchange; strong and specific bonding. #### Surface Chelation: - Chelate-ring structure formed when two or more ligands bond to a metal cation. - Important in strongly binding many micronutrients into organic matter (e.g., Fe³⁺, Cu²⁺, Zn²⁺, Mn²⁺). - Very slow release. #### Dissolution/Precipitation: - Ions dissociate from solid, go into solution. - Example: $$\text{CaCO}_3 \rightarrow \text{Ca}^{2+} + \text{CO}_3^{2-}$$ ### Electroneutrality - An important concept: all solutions (and systems) must be electrically neutral. - $$\sum z_i c_i = 0$$ - $c_i$ = concentration of ionic solute - $z_i$ = charge - Example: $$2[\text{Ca}^{2+}] + 2[\text{Mg}^{2+}] + 2[\text{Na}^{+}] + 2[\text{K}^{+}] + [\text{H}^{+}] =$$ $$2[\text{SO}_4^{2-}] + [\text{NO}_3^{-}] + [\text{Cl}^{-}] + [\text{HCO}_3^{-}] + 2[\text{CO}_3^{2-}] + n[\text{An}^{-}] + [\text{OH}^{-}]$$ ### Soil pH - **Acid solutions:** pH 7.0 - **Acid:** Proton donor, increases hydronium ion concentration. - **Base:** Proton acceptor, reduces hydronium ion concentration (increases hydroxide ion concentration). #### Soil pH is pH of soil solution. - **Slightly acid:** 6.0 – 6.6 - **Moderately acid:** 5.0 – 6.0 - **Strongly acid:** 9.0 #### Active Acidity: - Due to H⁺ ion activity in the soil solution at any given time. #### Reserve Acidity: - Represented by H⁺ and Al³⁺ that are easily exchanged by other cations. ### Sources of Acidity in Soil - Hydrogen and Aluminium cations are responsible for soil acidity. - **Exchangeable Hydrogen** is the main source of acidity at pH 6 and above. - Below pH 6, **Aluminium** is the main source of acidity due to dissociation of Al from clay minerals. - $$\text{Al}^{3+} + \text{H}_2\text{O} \rightarrow \text{Al(OH)}^{2+} + \text{H}^+$$ - $$\text{Al(OH)}^{2+} + \text{H}_2\text{O} \rightarrow \text{Al(OH)}_2^{+} + \text{H}^+$$ - $$\text{Al(OH)}_2^{+} + \text{H}_2\text{O} \rightarrow \text{Al(OH)}_3 + \text{H}^+$$ 1. **Nitrification:** Ammonium to Nitrate (oxidation of NH₄⁺) - $$\text{NH}_4^+ + 2\text{O}_2 \rightarrow \text{NO}_3^- + \text{H}_2\text{O} + 2\text{H}^+$$ 2. **O.M. decomposition:** - Organic acids ionized: $$\text{R-COOH} \rightarrow \text{R-COO}^- + \text{H}^+$$ - Respiration: $$\text{CO}_2 + \text{H}_2\text{O} \rightarrow \text{H}_2\text{CO}_3 \rightarrow \text{H}^+ + \text{HCO}_3^-$$ ### Acid Rain - Caused by burning fossil fuels. - Burning oil, gas, and coal in power stations releases Sulfur dioxide (SO₂) into atmosphere. - Burning oil and gasoline in motor vehicles releases nitrogen oxides (NOₓ) into atmosphere. - These gases mix with water droplets, creating weak solutions of nitric and sulfuric acids, which fall as acid rain. - $$\text{SO}_2 + 2\text{OH} \rightarrow \text{H}_2\text{SO}_4 \rightarrow \text{SO}_4^{2-} + 2\text{H}^+$$ - $$\text{NO}_2 + \text{OH}^- \rightarrow \text{HNO}_3 \rightarrow \text{NO}_3^- + \text{H}^+$$ ### Soil Pollution/Contamination - Occurs when hazardous chemicals are buried, spilled, or migrate into uncontaminated soil (improper disposal, pesticides/fertilizers, industrial processes, contaminated water, industrial smokestacks). - **Common chemicals:** Petroleum hydrocarbons, polynuclear aromatic hydrocarbons (naphthalene), solvents, pesticides, lead, heavy metals. - Correlated with industrialization and chemical substance intensity. - Concerns stem from health risks (direct contact, vapor, secondary contamination of groundwater). #### Pollutant interaction with soil depends on: 1. Type and physicochemical state of pollutant. 2. Type and physicochemical state of the soil. 3. Type and state of microorganisms. 4. External factors (temperature, nutrients, oxygen). #### Physical forms of organic pollutants in soil: 1. Solid particles as a separate phase. 2. Liquid film covering soil particle. 3. Adsorbed onto surface of soil particle. 4. Absorbed into soil particle (incorporated into mineral lattice). 5. In soil macropores in water phase (dissolved). 6. In soil micropores in solid or liquid form as a separate phase. ### Effects of Soil Pollution #### Health effects: - Direct contact, inhalation of vaporized contaminants. - Infiltration into groundwater aquifers used for human consumption. - Can lead to pollution-related diseases. - Chronic exposure to chromium, lead, other metals, petroleum, solvents, pesticides, and herbicides can be carcinogenic, cause congenital disorders, or other chronic conditions. - Naturally occurring substances (nitrate, ammonia from livestock manure) also identified as health hazards. #### Ecosystem effects: - Soil chemistry changes can alter metabolism of microorganisms and arthropods. - Can lead to eradication of primary food chain, affecting predator/consumer species. - Contaminants alter plant metabolism, reducing crop yields. - Reduced crop cover affects soil conservation and increases erosion. ### Element Associations with Different Soil Fractions - **Colloid:** A substance microscopically dispersed evenly throughout another substance, consisting of a dispersed phase and a continuous phase. - **Inorganic colloids** (clay minerals, Fe-Mn hydrous oxides): - Usually make up bulk of soil colloids. - Particles less than 0.001 mm in size (clay fraction ### Remediation - Act of correcting an error or fault. - **Objective:** Reduce risks to human health, environment, and property to acceptable levels by removing/reducing contamination or blocking exposure pathways. - **Remediation of soils polluted with radionuclides depends on:** - Chemical forms of the element. - Characteristics and concentrations. - Effects on environmental and human health. - **Three basic options for remedial actions:** monitored nonintervention, containment, removal. #### Remediation Techniques: | Methods | In situ | Ex situ (on/off site) | |:-----------------|:------------------------------|:--------------------------------------| | **Thermal** | Thermal induced SVE | Combustion, Pyrolysis, Thermal desorption | | **Physical-chemical** | Air injection, Soil washing, Leaching, Vapor extraction, Solidification/stabilization, Electrokinetic | Desorption in reactor, Chemical extraction, Extraction with solvents, Soil washing, Electrokinetic/Electrodialytic | | **Biological** | Bioventing, Phytoremediation, Monitored Natural Attenuation, Induced biodegradation | Landfarming, Bioreactor, Landfarming | | **Special processes** | Phase recovery | | ### Bioremediation - Waste management technique using organisms to remove or neutralize pollutants. - EPA definition: "treatment that uses naturally occurring organisms to break down hazardous substances into less toxic or non toxic substances". - Technologies classified as in situ or ex situ. ### Phytoremediation - Treatment of environmental problems using plants to mitigate contamination without excavating and disposing of material. - Cost-effective, plant-based approach. - Plants concentrate elements/compounds and metabolize molecules in tissues. - Targets: toxic heavy metals, organic pollutants. - Examples: Restoration of abandoned metal mine sites. #### Hyperaccumulators: - Plants that concentrate pollutants to a minimum percentage (e.g., >1000 mg/kg for Ni, Cu, Co, Pb; >10,000 mg/kg for Zn, Mn). - Capacity due to hypertolerance/phytotolerance (adaptive evolution). - Examples: Indian mustard (Cd, Cr, Cu, Ni, Pb, Zn), sunflower (Pb, U, Cs-137, Sr-90). #### Advantages: - Plants easily monitored. - Possibility of recovery and reuse of valuable metals. - Least harmful method, uses naturally occurring plants, preserves environment. #### Limitations: - Limited to surface area and depth occupied by roots. - Slow growth, requires long-term commitment. - Not possible to completely prevent leaching into groundwater. - Survival of plants affected by toxicity and general soil condition. - Bio-accumulation requires safe disposal of affected plant material. ### Bioremediators - Microorganisms used for bioremediation. - Heavy metals in harvested biomass can be concentrated by incineration or recycled. - Example: *Deinococcus radiodurans* (most radioresistant organism) modified to consume and digest toluene and ionic mercury from radioactive nuclear waste. - **Mycoremediation:** Form of bioremediation using fungi to decontaminate areas. ### Quality Assurance (QA) - System of activities to provide assurance that a product meets quality standards with a stated level of confidence. #### Quality Assurance Consists Of: - **Quality Control (QC):** Controls product quality and maintains the system in statistical control. - **Quality Assessment:** Assesses overall precision and accuracy of data after analyses; ensures QC is effective. - **Quality Assurance (Broad Plan):** Maintains quality in all program aspects; includes documentation, training, study design, data management, and QC measures. ### Quality Assessment Techniques #### Internal: - Repetitive measurements - Internal test samples - Control charts - Interchange of operators - Interchange of equipment - Independent measurements - Definitive method measurements - Audits #### External: - Collaborative tests - Exchange of samples - External reference materials - Standard reference materials - Audits #### Principle: Plan, Organise, Execute, Monitor (POEM) ### Basic Elements of Quality Control - Technical competence of staff. - Suitable facilities and equipment. - Good laboratory practices. - Good measurement practices. - **Standard Operating Procedures (SOPs):** Specify how operations should be done; provide basis for data comparability among laboratories. - Protocols for specific purposes. - Inspection. - Documentation. - Training. ### Data Precision and Verification - **Blind Spikes:** Samples spiked with known amounts of radionuclides or nonradioactive substances; quantity known to sampling organization but not analytical lab. - **Duplicate Sampling within Organizations:** - *Duplicate sample:* Two samples from same location at same time. - *Split sample:* Single sample divided into two portions for separate analysis. - **Duplicate Sampling between Organizations:** Comparing data collected simultaneously by different organizations. ### Tests for Internal Consistency of Data #### Data Plots: - **Dotplot / Scatterplot:** Data plotted as is. #### Boxplot: - No distributional assumptions; no estimate of mean or S.D. - **Potential outliers:** Points outside $q_1 - (1.5 \times \text{IQR})$ and $q_3 + (1.5 \times \text{IQR})$. - **Problematic outliers:** Points outside $q_1 - (3 \times \text{IQR})$ and $q_3 + (3 \times \text{IQR})$. #### Dixon test: - $$R = \frac{X_n - X_{n-1}}{X_n - X_1}$$ - $X_n$ = Highest value, $X_{n-1}$ = Next higher value, $X_1$ = lowest value. #### Grubbs test: - $$T = \frac{X_n - \mu}{s}$$ - $X_n$ = Highest value, $\mu$ = mean value, $s$ = S.D. - If data fails these tests at a significance level of $\alpha = 0.05$, they are considered **outliers**. - **Outlier:** An observation abnormally distant from other values in a random sample. ### Quality Control Charts - Take samples of a certain size. - Produce line charts of variability in samples. - Upper and lower control limits fixed at $\pm 3\sigma$ limits. - If trend emerges or samples fall outside limits, process is out of control; action taken to find cause. - **X-bar chart:** Sample means plotted to control mean value of a variable. - **R chart:** Sample ranges plotted to control variability of a variable.