**Abercrombie formula** |
This is the most commonly used counting correction formula. It attempts to correct for over counting by adjusting the count by a factor based on the section thickness and the average height of the particles being counted. This is a biased formula and should be avoided. |

**accuracy** |
Accuracy is a measure of how close an estimate or measurement is to the actual or true value. |

**anisotropic** |
Having properties that differ according to the direction of measurement. |

**antithetic variates** |
A method reducing the variance. The idea is to tie together things such that an increase in variance in one place reduces the variance in another. The effect is to reduce the overall variance. |

**artificial edges** |
Objects have natural edges. Cutting objects such as organs produces artificial edges. Care must be taken when probes cross-artificial edges. |

**associated point method** |
A method of deciding if an object is to be counted by a counting frame. In this method, a particle is reduced to a unique point. This unique point is then associated with the object. For example, this point may be the nucleus, center of mass, the leftmost point or the topmost point. Whatever type of point is used, it is critical that this point can be uniquely identified for all of the objects to be counted. If this unique point falls within the counting frame, then the object is counted in that counting frame. It may be problematic to apply the associated point rule in some cases because of the difficulty in deciding whether the associated point falls inside or outside the sampling volume in marginal cases. Note: there are at least two methods of deciding if an object is to be counted by a counting frame. The **forbidden line method** is the other method of particle counting. |

**area sampling fraction** |
The ratio of the counting frame area to the area formed by the fractionator sampling grid. Often abbreviated as *asf*. |

**average** |
See **mean.** |

**bias** |
A statistical sampling or measurement error caused by systematically favoring some outcomes over others. Bias causes the mean of the estimated values to deviate from the true value. Some common sources of bias are 1) limitations of technique (inadequate contrast, positive section thickness, overlap and truncation effects etc.) 2) statistical bias arising from the sampling design, 3) geometrical bias arising from discrepancies between a stereological model and reality. |

**boundary** |
The outline of a profile. A boundary is the outer edge of a 2D area. A boundary can be formed by the perimeter of a profile resulting when a plane intersects a 3D object. |

**cascade design** |
In a cascade design the material being sampled is repeatedly subdivided. Only a fraction of the material advances to the next step of the sampling procedure. The final estimate is multiplied by the inverse of all of the fractions to estimate the quantity that applies to the original specimen. |

**Cavalieri estimator** |
An unbiased stereological method that estimates the volume of a structure from individual parallel cross-sectional areas (typically sections) employing Cavalieri principle. Typically, the areas are estimated using a point-counting method. |

**Cavalieri principle** |
The Cavalieri principle states that the volume of two objects is the same if planes parallel to the base of the two objects intersects the objects to form profiles that have equal areas. |

**coefficient of error (CE)** |
The standard deviation of the sample divided by the mean of the sample. This value reflects on the variability of the estimated mean with respect to the population mean. The CE and the CV have similar definitions. The CV is a parameter while the CE is a statistic. Statistics books use CV as either a parameter or statistic. In stereology, the use is differentiated by the use of these two terms. |

**coefficient of variance (CV)** |
The standard deviation of the population divided by the population mean. The coefficient of variation is abbreviated as CV. This reflects on the variability of the population. Statistics usually uses CV to mean both a parameter and a statistic. In stereology, the uses are differentiated by using CE for the statistic and CV for the parameter. |

**co-latitude** |
The angle between the z-axis and a selected direction. It must be a value in the closed interval 0 to pi. The 0 angle is the angle parallel to the positive z-axis. The angle pi is the angle parallel to the negative z-axis. An angle of pi/2 defines the points on the equator. |

**consistent estimator** |
An estimator which may be biased, but approaches the true value as the sample size increases. When the sample size is the population, the estimate is unbiased. A ratio estimator such as volume fraction is often a consistent estimator. Note that a consistent estimator and an unbiased estimator are not equivalent. |

**counting frame** |
A two-dimensional stereological probe that is used with specific counting rules to count or select particles. The counting frame is a rectangle with extensions of two infinite rays. It is typically displayed using the colors red and green, which assist in implementing the counting rules. Use of a counting frame along with the counting rules results in all particles having an equal probability of being selected, regardless of shape, size, orientation, and distribution. |

**counting space** |
The 3D volume contained within the Optical Disector probe. It is calculated by the counting frame area multiplied by the height of the Optical Disector. |

**cut section thickness** |
The section thickness as measured directly from the sectioning device (cryostat, microtome, etc.). This is the thickness of a section prior to histological processing, which may cause shrinkage in the z-axis. Also known as the block advance of the microtome. |

**cycloid** |
A sine weighted curve that is used in some stereological probes in vertically sectioned material. Normally the curve is described by its parametric equations, which are: x = a(t-sin t), y = a(1-cos t) The easiest way to think of a cycloid is to imagine the path taken by a point on a circle rolling a straight line. The value a in the equations is a scale factor. The maximum height, the y value, is 2a. The length of the curve from the low point to the high point is 8a. The horizontal distance from the low point to the high point is 2pi a. The cycloid is important in stereology because of a property sometimes referred to as the sin-weighted or cos-weighted property of the curve. The sine of the co-latitude is proportional to the length of the curve parallel to the co-latitude angle. |

**cytoarchitecture** |
The arrangement of cells in the brain, especially the cerebral cortex. |

**Delesse principle** |
The Delesse principle states that the ratio of the area occupied by a component relative to the entire profile area is a consistent estimate of the volume fraction of the component in the object. More simply, the area fraction is equal to the volume fraction. The Delesse principle is written as: AA =VV |

**design based stereology** |
An unbiased stereological technique using sampling design. This method eliminates the need for making assumptions about the size, shape, or orientation of objects. Prior to the development of these methods, model-based approaches were used which made assumptions about the nature of the objects being studied, such as their shape, orientation, or distribution. Design-based methods have reduced the assumptions required so as to be practically assumption free. |

**Disector** |
A stereological probe for counting or selecting objects using a pair of adjacent physical sections. The method uses a counting frame with counting rules to determine the objects that are counted. The Disector is also referred to as the Physical Disector, especially in literature that follows the introduction of the Optical Disector. The term comes from the composition of the terms *di* for two and *section *the disector uses two sections. The two sections must be close enough that it is possible to infer what lies between the two sections. This makes it possible to use the disector to sample volume. |

**double disector** |
The double disector is a means of estimating the number of small objects in a two-step disector process. The first step is to estimate the number of large objects such as cells. The second step is to estimate the number of small objects to large objects. The procedure ends up canceling out the section thickness, which is often difficult to determine in very thin sections. |

**efficiency** |
The efficiency of a stereological method is defined as the measure of the precision it offers per unit cost; the precision of an estimate is proportional to the reciprocal of its error variance. An inefficient method is one that obtains a poor result after much work. In statistics, a sampling method is said to be more efficient than another is if the variance is less than another is and the cost of collecting the samples is the same. |

**empirical distribution function plot** |
A graphical representation of the proportion (or frequency) of values less than or equal to each value. |

**estimate** |
A well-defined numerical value that approximates a quantifiable parameter. In stereology, an estimate is generated by an estimator. Estimates are used when it is not feasible (or necessary) to obtain an exact measurement. |

**estimator** |
An estimator is a formula that generates an estimate. The inputs to an estimator are measurements from samples and the result is an estimate of a quantifiable parameter. Examples of estimators are the Cavalieri estimator and the Optical Fractionator. |

**exclusion lines** |
One of a set of lines that comprise a counting frame. The counting frame is created from 2 sets of lines at right angles to create a rectangle. If an object crosses the set of lines designated as exclusion lines, the object is not counted. Exclusion lines are typically displayed as red or dashed lines. |

**expected value** |
The mean of a random variable that has a probability distribution function. The expected value is defined for discrete values and for continuous values. If all values have an equal probability, then the expected value is the same as the mean. |

**finagler constant** |
A constant, which cannot be tested, but which, if taken as faith, will make experimental data come out the way you would like to have it. This is sometimes referred to as a correction factor. Design-based stereology avoids the use of these finagling constants. (The word finagle means to achieve by trickery.) |

**fakir probe** |
A probe composed of parallel lines. The lines are used to sample surfaces. The probe gets its name from the fakir bed of nails. |

**final magnification** |
Final magnification is a unitless value that expresses the ratio of the size of an image and the size of the original object. The final magnification is the product of all of the magnifications used in the imaging system. |

**Floderus formula** |
The Floderus formula is a model-based approach to estimating numerical density of particles. This formula appeared at approximately the same time as the **Abercrombie formula**. The formula relates the numerical area density of particles to the numerical volume density along with a correction term. This is a model-based formula and should be avoided unless you are confident it models your data. For details, see the Floderus section. |

**Forbidden line method** |
Counting method used with objects that have a profile larger than a single point. The rules are applied to the counting frame and decide whether a particle should be counted depending on how a particle intersects or does not intersect the lines forming the counting frame. If a particle falls inside of the counting frame, but does not cross a forbidden line it is counted in this counting frame. If a particle touches one of the forbidden lines, then it is not counted in this counting frame. The details of the counting rules can be found in the section on counting rules. |

**fractionator** |
A systematic random sampling method that selects a portion of a region of interest. The fractionator principle is used in many areas of design-based stereology. Fractionator based sampling schemes are unbiased. |

**general requirement** |
The requirement that the observer understands what they are observing. Being able to identify what is, and what is not, a member of a population is important. This is one of the basic assumptions made in almost all research. For stereologists the issue is extremely important since the materials are cut into sections. Being able to understand what the profiles mean and how they are possibly connected is necessary to obtain unbiased results. |

**geometric mean** |
The geometric mean differs from the arithmetic mean by multiplying all numbers together and then taking the (1/n) root of the product. |

**geometric probe** |
A geometric shape, typically a set of lines, points, cycloids, or curves, which is overlaid on an object to investigate and obtain quantitative information of the object. |

**geometrical bias** |
Bias introduced into the implementation of a stereological estimator by differences between the theoretical design and the actual implementation. |

**graphical unfolding** |
A model based estimation technique for objects that are a solid of revolution. |

**gray level index method** |
A methodology for preprocessing an image to automatically identify boundaries between cytoarchitectonically defined brain regions. |

**guard zone** |
A zone used to avoid sampling near the upper and lower section edges, places where the material may be altered due to the histological sectioning process. Optical Disectors are used with guard zones to avoid issues with lost caps. |

**Holmes effect** |
An overprojection issue due to the positive thickness of a section. A nontransparent object embedded in a section is seen as large as or larger than it should be due to the thickness of a section greater than. |

**Horvitz-Thompson estimator** |
An estimator that is based on knowing the probability of selecting a member of a population. |

**inclusion lines** |
One of a set of lines that comprise a counting frame. The counting frame is created from 2 sets of lines at right angles to create a rectangle. If an object crosses the set of lines designated as inclusion lines, or lies within the counting frame the object is counted. Inclusion lines are typically displayed as green or solid lines. |

**intensity** |
The intensity is related to the density of the probe. The more intense the probe, the more dense the geometries. The more intense the probe the greater the number of counts as the probe is more likely to intersect the objects of interest. |

**isector** |
Stereological procedure to create IUR sections or slices from small samples. The object is imbedded in a sphere and randomly rolled. Sections or slices cut from the randomly rolled isector are IUR. Creating large spheres for large objects is often prohibitive. Use the** orientator **for large objects. |

**isotropic** |
The property of being identical in all directions. |

**isotropic rotator** |
The Isotropic Rotator is the version of the Planar Rotator used with** Isotropic Uniform Random (IUR) sections**. |

**isotropic section** |
The section has been taken using an IUR direction. These sections are also known as IUR sections. This is a completely random section in which all possible directions are possible. The IUR direction is the direction perpendicular to the face of the isotropic section. Such sections can be produced with the isector or the orientator. |

**isotropy** |
The condition where the orientation of the probe does not affect the mean. Another way of looking at this is that the property appears to be the same independent of the orientation in which it is viewed. |

**Isotropic Uniform Random (IUR) sections** |
Sections cut in a manner that they fulfill the criteria to be both isotropic (having no preferred orientation) and a uniformly distance (interval) apart. The orientation of the sections is selected at random. Note, the sectioning methods used by most researchers, i.e., sagittal, coronal, etc., have preferred orientations and do not result in IUR sections. |

**IUR sections** |
See Isotropic Uniform Random (IUR) sections. |

**Köhler illumination** |
A method of microscopy that provides optimum specimen visualization using brightfield illumination by focusing the light from the microscope condenser at the level of the specimen. |

**latitude** |
The part of a 3-dimensional direction that is given as the angle between the x-axis and the x-y plane projection of a selected direction. Latitude is usually given as an angle in the range of 0 to 2 pi or in the range of -pi-to-pi. An angle of 0 corresponds to the direction parallel to the positive x-axis. |

**lineal structure** |
A lineal structure is an object that has a length that is much greater than its diameter. Dendritic processes capillaries are examples of lineal structures. Theoretically, a lineal structure is a 1-dimensional object, but in practice, this is not the case. The length or length density of lineal structures is often of interest in stereology. Note, as the ratio of the diameter to the length increases, determining length becomes more difficult. |

**local stereology** |
Those sampling methods that are based on using unique and arbitrary points inside of particles. Examples are the nucleator, planar rotator, and the surfactor. |

**lookup section** |
The disector method requires two sections. One is called the reference section and the other is called the lookup section. Particles are selected in the reference section and counting is done in the lookup section. |

**mean** |
The arithmetic average of two or more values. This is defined as the sum of all of the values divided by the number of values. Placing a bar over the variable often denotes the mean. The mean can be determined for both discrete data and continuous functions. The mean is often used to represent a set of values. Other statistics used to describe the central value are the mode and median. |

**median** |
The median value is the middle value. If the values are sorted in either ascending or descending order then the value in the middle of the list is the median value. Other statistics used to describe the central value are the mean and mode. |

**mensuration** |
The science of measurement. The science studies how to represent geometrical quantities numerically. |

**Merz** |
A stereological probe designed to estimate the length of objects. |

**metrology** |
The study of measurement involves measuring quantities, calibration of instruments, and determining the uncertainty of a measurement. |

**mode** |
The mode is the value that is the most common in a list of values. The mode is easy to pick out of a histogram. It is the highest point in a histogram. For example, if the list of values is 1, 2, 2, 2, 3, 3, and 4, the mode is 2 since there are more 2’s than any other value. Other statistics used to describe the central value are the mean and median. |

**model-based stereology** |
Stereological methods, which assume that objects have particular size, shape, and orientation which can be approximated using a mathematical model (such as the method of Abercrombie, 1946). Model based methods are biased unless the objects exactly match the model. Until the advent of design-based stereology, model based stereology was the only form of stereology. In general, model-based methods are biased unless the objects exactly match the model. This is unlikely and usually cannot be determined. |

**Mounted section thickness** |
The thickness of tissue sections after histological processing. |

**morphometry** |
The measurement of the shape of objects. Morphometry includes a large range of measurements including numbers, length, surface area, volume, angles, and curvature. There are also measurements of shape ratios such as how round or box-like an object is.. Distributions and textures can also be measured. |

**multi-level design** |
See **cascade design.** |

**Nearest-Neighbor Distance Distribution Function (NNDDF)** |
A relative frequency distribution of the location of cells in a given volume. |

**nomogram** |
A graphical means of solving an equation. The 2 dimensional drawing allows for quick approximations. Nomograms are often used in stereology to make predictions about how much sampling is required to obtain a given CE. The nomogram makes it possible to explore a number of situations in a short period of time. |

**Nucleator** |
A design-based local estimator that uses the intersection of rays with the cell surface for the estimation of the volume and cross-sectional area of cells or small objects (nuclei, etc.). |

**nugget effect** |
A measure of the variance of an estimator within a section or slice. The nugget effect is now called the variance due to noise. |

**number weighted** |
This is the same as the mean. By explicitly stating that the mean is weighted by the number of particles simply serves to distinguish the number weighted mean from the volume-weighted mean. |

**Number-weighted mean volumes (arithmetic mean volume)** |
Generally known as the mean volume, unambiguously stating that the mean is weighted by the number of particles rather than the volume of the particles (volume weighted mean). |

**observational bias** |
Bias due to the observer being unable to clearly observe what is necessary. Observational bias may be due to poor staining or the inability to determine if a particle is a member of the population or not. Observational bias may be due to problems in determining the boundaries of an object (possibly due to low or poor contrast) or whether or not a particle is in focus. Section thickness leads to the** Holmes effect**. Overlapping particles may be difficult to separate or determine their extent. Although a technique may be unbiased in practice, it may be biased by observational difficulties. |

**Optical Disector** |
A stereological probe for counting or selecting objects in a tissue section. This is an extension to the basic Disector method, which is applied to a thick section using a series, or stack, of Disectors. Rather than using pairs of physical sections (the basic Disector method), optical sectioning is used by creating focal planes with a thin depth-of-field through the section. The Optical Disector begins with a lookup section at the top of the optical disector and ends with a reference section at the bottom of the optical disector. The focal plane is the current reference section. The lookup section is immediately above the focal plane. A particle in focus at the top of the optical disector is therefore seen in the lookup section and not counted. A particle in focus at the bottom of the optical disector is in the reference section and therefore not in the lookup section, is counted. Counting frame rules are applied when the particle first comes into focus. |

**Optical Fractionator** |
A design-based stereological method using a two-stage systematic sampling method that is used to estimate the number of objects in a specified region of an organ. This method combines the Optical Disector method with the fractionator sampling method. Like the Physical Fractionator, the Optical Fractionator is typically used when the population is too large to count exhaustively. |

**Optical Rotator** |
The Optical Rotator is a local stereological estimator that can estimate volume of objects in both isotropic and vertical sections. The surface area of objects can also be determined if the sections are isotropic sections. The estimator consists of a stack of parallel test lines that are rotated 90 degrees to each other at successive z values. Other local estimators are the Planar Rotator, the Nucleator, and Surfactor. |

**orientator** |
Stereological procedure to create IUR sections or slices from large samples. The orientator begins by selecting a pair of angles in such a manner that the direction described by the pair is a uniformly random direction. Once the angles have been chosen a slicing technique can be used to slice the specimen into sections that are perpendicular to this direction. |

**ortrip** |
The term ortrip is formed from the words “orthogonal triplet”. Measurements are taken from 3 mutually perpendicular IUR sections instead of 3 independent IUR sections. Theory shows that the precision of estimates from ortrips is better because the variance is reduced. Estimates on ortrips have been shown to exhibit antithetic variances. |

**overprojection** |
Overprojection is the process of obtaining an estimate that is too large. Overprojection can be due to a number of problems including systematic, and observational biases. For example, an inability to separate the top from bottom surfaces can lead to an overprojection of volume. |

**Pappus Theorem** |
This theorem states that a planar figure rotated about an axis describes a volume of revolution that is equal to the planar area of the figure times the length of the path traveled by the centroid of the figure. |

**parameter** |
A parameter is a quantitative value of a population. Examples are the number of cells in a brain, or the amount of quartz in a granite. Estimates of parameters can be inferred from samples. |

**particle** |
Stereologists uses the term particle instead of cell or grain to represent objects of interest. Instead of counting cells, or estimating grain size, a stereologist counts particles and estimates particle size. The mathematical definition is a compact and connected subset of R3 with a piecewise smooth boundary. |

**physical disector** |
A stereological method for counting or selecting objects using a pair of adjacent physical sections. The method uses a counting frame with counting rules to determine the objects that are counted. The Physical Disector is also often referred to as the Disector, especially in literature that precedes the introduction of the Optical Disector. To make matters even more confusing, often the term Physical Disector is used in place of the term Physical Fractionator. |

**Physical ****Fractionator** |
A design-based stereological method using a two-stage systematic sampling method that is used to estimate the number of objects in a specified region of an organ. This method combines the Physical Disector method with the fractionator sampling method. Like the Optical Fractionator, the Physical Fractionator is typically used when the population is too large to count exhaustively. Since the Physical Fractionator relies on the Physical Disector method, pairs of thin sections are required. |

**Planar Rotator** |
A method used to estimate the volumes of objects. Before the Optical Rotator was published, the Planar Rotator was called the Rotator. This method overlays a grid on the object. The grid has a number of parallel lines that are equally spaced. The intersections between the lines and boundaries of the object are identified. The Planar Rotator can be used with both vertical and isotropic sections. When used with vertical sections the probe is called the Vertical Rotator. When used with isotropic sections the probe is called the isotropic rotator. |

**planimeter** |
A device that is used to measure areas. The device has an arm that can be extended and rotated to trace the region of interest. Originally, planimeters were mechanical devices. |

**point sampled intercepts** |
A method for efficiently estimating the volume-weighted mean volume of cells or small objects. The particles are sampled with a point. This leads to the volume weighting of the results. After selection by the point, a line is used to measure the particle. |

**precision** |
A measure of how close a number of estimates or measurements are to each other. If an estimator is precise, the estimates cluster. |

**probe** |
A geometric shape, typically a set of lines, points, cycloids, or curves, which is overlaid on an object to investigate and obtain quantitative information of the object. |

**profile area** |
The area seen on a cut surface. When an object, such as a cell is cut, the profile of the cross-section is visible. A profile is formed by slicing physically or virtually through material. |

**profile counting** |
The act of counting the cross-sections of an object rather than the entire object. When an object, such as a cell is cut, the profile of the cross-section is visible. Objects counted using a profile counting method in a single 2D representative section are susceptible to sampling bias. |

**quadrat** |
A sample that has area. Quadrats are normally rectangular although they may be any shape. A photograph or digital image can be a quadrat. A view through the oculars of a microscope may be a quadrat. The objects in the quadrat are sampled with probes. In macroscopic work such as ecological studies, the quadrat may be mapped out and staked. |

**random** |
Random is unpredictable. There is some element of chance. This is the opposite of deterministic, in which the next number or event is knowable. |

**randomized start** |
A random initial section is chosen within the sampling interval to begin the section sampling using a random number generator. The use of a randomized start is critical to systematic random sampling. |

**reference section** |
The Physical Disector uses a technique that employs two sections. The two sections are called the reference section and the lookup section. These sections must be adjacent sections. The counting rule for the Physical Disector states than a particle is counted only if it appears in the reference section, but does not appear in the lookup section. |

**reference trap** |
A phenomenon that occurs when reporting the density of objects. Normally, people assume that when the density increases it is due to an increase in the number of objects. However, the density also can increase if the number of objects remains constant and the volume of the region (known as the reference volume) shrinks. Since this assumption has often been made, it is now a well-known trap that investigators may fall into. One of the advantages of the Optical Fractionator is that it is not subject to this trap. |

**reference volume** |
The volume of the region of interest. Stereological work may result in a density with quantity per unit volume. The density should be multiplied by the reference volume to determine an absolute quantity. This is done to avoid the reference trap. It is important to know the reference volume to report the correct population densities. This is done to avoid the reference trap. |

**representative section** |
Acting as a true example of an entire region. In reality, upon close inspection sections that initially appear to be â€œrepresentative sectionsâ€ often are not actually representative of the entire region. |

**rotator** |
A method for estimating the mean number-weighted particle volume, similar to the Nucleator. Also referred to as the **Planar Rotator.** |

**second order stereology** |
Methods that deal with the relationship among objects, such as the distribution. |

**section** |
Stereologists use the term section differently than biologists. Stereologists define section as a cut through material that has effectively zero thickness compared to the size of the particles being studied. Biologists refer to sections as thick slices through tissue. The actual thickness of sections can leads to the** Holmes effect. ** |

**section sampling fraction** |
The known interval of sections sampled through an object of interest (e.g., examining 1 in every 5 sections has a section sampling fraction of 1/5). Often abbreviated as *ssf*. |

**selector** |
Estimator used to count particles. The innovation in this estimator was being able to avoid determining the section thickness. This estimator is no longer in use. |

**sieving distributions** |
The concept that objects are separated into distributions based on some characteristic such as volume. A large screen removes the largest pieces. A finer screen removes the next sized particles. Eventually the finest screen separates out the smallest particles. These sorts of distributions are often important in understanding the makeup of a population. |

**shrinkage** |
The difference between the original section thickness (section cut thickness) and that after it has been histologically prepared for stereology (mounted section thickness). |

**slab** |
A very thick piece through an object. A slab is thick enough that it is not possible to view completely through the material being studied. |

**slice** |
A slice is a thick piece cut from the original material. A slice is thick relative to objects being sampled. It must be possible to view throughout the slice, but the objects being studied must have a smaller height than the slice’s thickness. The biologist’s use of the term ‘section’ is similar to the stereologist’s term ‘slice’. |

**Space Balls** |
A design-based stereological method that estimates length of tubular objects such as dendrites or blood vessels in thick sections. The method works by counting the number of transections of the objects to be counted with a virtual sphere. Also known as Isotropic Virtual Spheres. |

**spatial grid** |
A stereological estimator that estimates surface area using three orthogonal fakir probes. The fakir probes are aligned with the principal axes. |

**spatial rotator** |
The spatial rotator is a local estimator that has not been published. It has been found to be difficult to use and is not in use. |

**standard deviation** |
The standard deviation is a measure of the spread of a set of values from the mean value. The units of the standard deviation are the same as the units of the original data making the standard deviation a linear value. The standard deviation is the square root of the variance. It is a measure of dispersion. |

**star volume** |
The star volume is that portion of space that can be seen from a particular position. For example, suppose that you are in a cave. The volume that can be illuminated by a flashlight is the star volume. The star volume is often used in interconnected spaces as a measure of volume. |

**statistic** |
A statistic is an estimate of a population parameter that is inferred from a sample. Usually the parameter is unknowable, but an estimate of the parameter is possible. Statistics are numerical properties of samples. For example, the a mean income for the people living in a particular country is a parameter. The mean income obtained by sampling a part of the population is a statistic. |

**stereology** |
The method of quantifying 2D and 3D structures using estimation methods. Sometimes stereology is described as the methods to estimate 3D quantities from 2D images, but this is not always the case. The term stereology was coined in 1962 at the first meeting of the ISS, the International Stereological Society. |

**stochastic geometry** |
The branch of mathematics that deals with random geometric structures. Many of the proofs in stereology are derived from this area of mathematical research. |

**stratify** |
To separate a population into mutually exclusive groups called strata. Each strata is considered separately when sampling. An example of stratification is to separate test animals based on gender. |

**surface vs. area** |
Stereology uses surface to indicate a 3-dimensional surface. Area refers to a 2-dimensional surface. For example, a cross section has area, but an organ has surface. The two are often confused. The original organ has surface, but once cut into slices, sections, or slabs, the profiles have area. The area is not the same as the surface. |

**surfactor** |
A stereological estimator that estimates surface area of convex particles by sampling the angle formed between an isotropic line and the boundary of the particle. The surfactor is rarely used in practice, but is efficient when performed by computerized systems. |

**systematic bias** |
An estimator is biased if the estimate it generates is biased. If the method fails to generate an unbiased estimate, then an unbiased estimate cannot be attained. Even if an estimator is not biased, bias can be caused by observational bias or geometrical bias. |

**Systematic Random Sampling** |
Regular sampling (using a known interval) that begins with a randomized start. SRS is the basis for design-based stereological sampling methods, such as the Fractionator. The term is abbreviated SRS or SURS. The U in SURS stands for uniform. |

**systematic sampling** |
Samples taken in a periodic and regular fashion. The periodicity may be distances in 1, 2, and 3 directions, in regular angular increments, or in any other periodic fashion. For example, taking every 12th slice of tissue. Some of the picks are i, i+12, i+24… |

**thickness sampling fraction** |
The proportion of the section height that is sampled. Calculated by dividing the mounted section thickness by the height of the counting space. It is also known as the height-sampling fraction. Often abbreviated as tsf. |

**total vertical projection****s** |
Estimator that estimates absolute length. The estimator requires that VUR slices are used. |

**trisector** |
A triple of vertical sections take equally spaced, i.e. 120° apart. The purpose of the trisector is to reduce the variance of the estimate. |

**true population** |
The actual total population of the region of interest. |

**Unbiased Counting Frame (UCF)** |
Also known as the counting frame. Compare with **counting frame**. |

**unbiased** |
The condition of having a bias that exactly equals 0. The lack of any systematic error. In this condition, the precision and accuracy are identical. It also means that the expected value is the true value. The term is often used in sense of unbiased for all practical purposes. If there is bias in the data, there is no way to detect it, filter it, or remove it from the data. |

**unbiased brick** |
An alternative probe to the Optical Disector that uses different counting rules. The counting brick requires that no object crosses the forbidden lines, not just the disector pair where the particle first comes into focus. The counting brick was developed by Howard and Reed. |

**underprojection** |
An underprojection is an estimate that is too small. An underprojection can be due to systematic or observational biases. The recognition of potential biases can make it possible to suspect biases that can cause an underprojection. For example, the possibility of not being able to identify all members of a population can lead to an underprojection. |

**variance** |
The variance is a measure of the dispersion of a set of values. The variance is the mean of the sum of the squares of the differences between the values and the mean of the sample. |

**variance due to noise** |
This was formerly known as the nugget effect. This is a measure of the variance from sampling within a section. This differs from the variance of systematic random sampling, which is due to differences between sections. |

**vertical rotator** |
The version of the Planar Rotator used with VUR sections. |

**vertical sectioning** |
A method of producing sections from a specimen that contains a natural flat surface, i.e., bone. A vertical axis is selected that is perpendicular to the surface. The material is randomly rotated about the vertical axis. Sections are then cut parallel to the vertical axis. |

**virtual counting space** |
The volume associated with the counting space or box. |

**volume weighted** |
A volume weighted result is one that has weights based on the volumes of the objects being studied. A volume-weighted sample is collected by using points to select particles. |

**volume-weighted mean ****volumes** |
The mean object volume if the objects are weighted (sampled) proportional to their volume. |

**Voronoi tessellation** |
Partitioning of plane using a set of points where in each cell partition, all points in the cell are closer to the point that defines the cell than to any other point. |

**VUR section** |
see **vertical sectioning.** |

**weighted mean** |
A weighted mean is often used in statistics. The weightings in stereology are usually number weighted and volume weighted. |