Developing a Meaningful Understanding of the Mean

2003 ◽  
Vol 9 (1) ◽  
pp. 22-26
Author(s):  
Elizabeth George Bremigan
Keyword(s):  
Arithmetic Mean ◽  
Statistical Concept ◽  
Statistical Measures ◽  
The Mean

One of the first statistical measures that students encounter in their study of mathematics is the arithmetic mean. The procedure for determining the arithmetic mean of a given set of numbers is relatively simple, because it requires only two computational skills: addition and division. Thus, students are often introduced to the arithmetic mean in grades 4 or 5. At this level, computation of the arithmetic mean is frequently presented as an application of division rather than as a statistical concept. Initially, the arithmetic mean is often called the average; in this article, the arithmetic mean will be referred to simply as the mean.

scholarly journals Comparative characteristics of the results of facial morphometry in patients with deep incisal overlap in the treatment of bracket systems and eliners

2020 ◽  
Vol 24 (2) ◽  
pp. 95-98
Author(s):  
S. Yu. Maksyukov ◽  
Nadezhda D. Pilipenko ◽  
K. D. Pilipenko
Keyword(s):  
Orthodontic Treatment ◽  
Arithmetic Mean ◽  
Morphometric Characteristics ◽  
Mandibular Angle ◽  
Deep Bite ◽  
Bite Angle ◽  
High Prevalence ◽  
Modern Methods ◽  
Quantitative Characteristics ◽  
The Mean

The relevance of studying the problem of deep incisal overlap (HF) among dentofacial anomalies (CCA) is due to the high prevalence of this pathology. Among modern methods of orthodontic treatment of pathology, the use of bracket systems and aligners is highlighted. The effectiveness of these techniques can be compared in determining the morphometric characteristics obtained by tele-radiography (TEG).Material and methods. The study involved 118 people with hydraulic fracturing, the average age was 38.7 8.5 years (64 women; 54 men). The first group consisted of 49 patients who underwent correction of AFA with eliners; second, 69 patients with bracket systems. To assess the effectiveness of treatment, TRG was performed. To present the results, in the case of quantitative characteristics, the arithmetic mean of the sample value (X) and the error of the mean (m) were calculated. For qualitative signs, the frequency of the sign (%) and its standard error (m%) were calculated.Results: The values ​​of the mandibular angle (G, ArGoMe), and the angles AB / ANS (AB / SpP), APg / ANS (MM), as well as the vertical dimensions of the jaws reached values ​​characteristic of an orthognathic bite. Angle increase SNB, NSL / ML4; angle reduction ANB.Output. Elimination of a deep bite is possible both with the use of bracket systems and aligners.

Estimation of plant biomass based on the allometric equation

1981 ◽  
Vol 11 (4) ◽  
pp. 833-834 ◽  
Author(s):  
David O. Yandle ◽  
Harry V. Wiant Jr.
Keyword(s):  
Linear Regression ◽  
Plant Biomass ◽  
Allometric Equation ◽  
Arithmetic Mean ◽  
Measurement Scale ◽  
Simple Linear Regression ◽  
The Mean ◽  
The One ◽  
Biased Estimates

Estimation of the parameters in the allometric equation by fitting a simple linear regression to the logarithmically transformed variables results in biased estimates of the arithmetic mean. This bias expressed as a percent of the mean approaches the limit −(1 − e−σ2/2) (100) as n increases. An adjusted estimator developed by Finney rather than the one given by Baskerville should be used when s2 is large and n is small. A change of measurement scale of the x or y variables presents no difficulty, but problems arise if variables are transformed to logarithms other than base e.

Hygienic Effects of Trimming and Washing Operations in a Beef-Carcass-Dressing Process†

1996 ◽  
Vol 59 (6) ◽  
pp. 666-669 ◽  
Author(s):  
C. O. GILL ◽  
M. BADONI ◽  
T. JONES
Keyword(s):  
Escherichia Coli ◽  
Standard Deviation ◽  
High Speed ◽  
Effective Means ◽  
Arithmetic Mean ◽  
Single Sample ◽  
Aerobic Bacteria ◽  
E Coli ◽  
Beef Carcasses ◽  
The Mean

Swab samples were obtained from the surfaces of randomly selected beef carcasses passing through a high-speed dressing process. A single sample was obtained from a randomly selected site on the surface of each selected carcass. Fifty such samples were collected at each of four stages in the process. The aerobic bacteria, coliforms, and Escherichia coli recovered from each sample were enumerated. Values for the mean log units and standard deviations of each set of 50 log values were calculated on the assumption that the log values were normally distributed. The log of the arithmetic mean was estimated from the mean log and standard deviation values for each set. The results show that the average numbers of E. coli, coliforms, and aerobic bacteria which are deposited on carcasses during skinning and evisceration are not reduced by trimming, and that washing approximately halves the average numbers of those bacteria on carcasses. It is concluded that commercial trimming and washing operations are not effective means of decontaminating beef carcasses.

On Gauss's proof of the normal law of errors

1933 ◽  
Vol 29 (2) ◽  
pp. 231-234 ◽  
Author(s):  
Harold Jeffreys
Keyword(s):  
Arithmetic Mean ◽  
Practical Case ◽  
Apparent Discrepancy ◽  
True Value ◽  
The Mean ◽  
Normal Law

Gauss gave a well-known proof that under certain conditions the postulate that the arithmetic mean of a number of measures is the most probable estimate of the true value, given the observations, implies the normal law of error. I found recently that in an important practical case the mean is the most probable value, although the normal law does not hold. I suggested an explanation of the apparent discrepancy, but it does not seem to be the true one in the case under consideration.

A note on the estimation of the mean of a homogeneous random field

1971 ◽  
Vol 8 (3) ◽  
pp. 626-629
Author(s):  
Michael Skalsky
Keyword(s):  
Finite Number ◽  
Random Field ◽  
Random Process ◽  
Mathematical Expectation ◽  
Covariance Function ◽  
Arithmetic Mean ◽  
Stationary Random Process ◽  
Homogeneous Random Field ◽  
The Mean

An important problem, arising in connection with the estimation of mathematical expectation of a homogeneous random field X(x1, ···, xn) in Rn by means of the arithmetic mean of observed values, is to determine the number of observations for which the variance of the estimate attains its minimum. Vilenkin [2] has shown, that in the case of a stationary random process X(x) such a finite number exists, provided that the covariance function satisfies certain conditions.

Assessment of the Hygienic Performance of a Sheep Carcass Dressing Process†

1998 ◽  
Vol 61 (3) ◽  
pp. 329-333 ◽  
Author(s):  
C.O. GILL ◽  
L.P. BAKER
Keyword(s):  
Escherichia Coli ◽  
Standard Deviation ◽  
Arithmetic Mean ◽  
Single Sample ◽  
E Coli ◽  
The Mean ◽  
Beef Carcass ◽  
Normally Distributed

Swab samples were obtained from the surfaces of randomly selected carcasses passing through a sheep carcass-dressing process. A single sample was obtained from a randomly selected site on the surface of each carcass. Twenty-five such samples were collected at each of four stages in the process. The aerobio bacteria, coliforms, and Escherichia coli recovered from each sample were enumerated. Values for the mean log and standard deviation of each set of 25 log10 values were calculated on the assumption that the log values were normally distributed. The log of the arithmetic mean was estimated from the mean log and standard deviation values for each set. The results showed that bacteria, including coliforms that were largely E. coli, were deposited in high numbers during skinning operations, mainly on the butts and shoulders of carcasses. The mean numbers of coliforms and E. coli on carcasses were little affected by eviscerating and trimming operations, although they were redistributed from the sites they occupied after skinning. Total counts were redistributed and augmented by eviscerating and trimming operations. Washing reduced the log numbers of all of the bacteria by approximately 0.5. The general hygienic characteristics of the sheep carcass dressing process were similar to those of a previously examined beef carcass-dressing process.

scholarly journals Characteristics of tropopause parameters as observed with GPS radio occultation

2014 ◽  
Vol 7 (11) ◽  
pp. 3947-3958 ◽  
Author(s):  
T. Rieckh ◽  
B. Scherllin-Pirscher ◽  
F. Ladstädter ◽  
U. Foelsche
Keyword(s):  
Annual Cycle ◽  
Radio Occultation ◽  
Southern Oscillation ◽  
Polar Vortex ◽  
Lapse Rate ◽  
Quasi Biennial Oscillation ◽  
Gps Radio Occultation ◽  
Statistical Measures ◽  
The Mean ◽  
Individual Profiles

Abstract. Characteristics of the lapse rate tropopause are analyzed globally for tropopause altitude and temperature using global positioning system (GPS) radio occultation (RO) data from late 2001 to the end of 2013. RO profiles feature high vertical resolution and excellent quality in the upper troposphere and lower stratosphere, which are key factors for tropopause determination, including multiple ones. RO data provide measurements globally and allow examination of both temporal and spatial tropopause characteristics based entirely on observational measurements. To investigate latitudinal and longitudinal tropopause characteristics, the mean annual cycle, and inter-annual variability, we use tropopauses from individual profiles as well as their statistical measures for zonal bands and 5° × 10° bins. The latitudinal structure of first tropopauses shows the well-known distribution with high (cold) tropical tropopauses and low (warm) extra-tropical tropopauses. In the transition zones (20 to 40° N/S), individual profiles reveal varying tropopause altitudes from less than 7 km to more than 17 km due to variability in the subtropical tropopause break. In this region, we also find multiple tropopauses throughout the year. Longitudinal variability is strongest at northern hemispheric mid latitudes and in the Asian monsoon region. The mean annual cycle features changes in amplitude and phase, depending on latitude. This is caused by different underlying physical processes (such as the Brewer–Dobson circulation – BDC) and atmospheric dynamics (such as the strong polar vortex in the southern hemispheric winter). Inter-annual anomalies of tropopause parameters show signatures of El Niño–Southern Oscillation (ENSO), the quasi–biennial oscillation (QBO), and the varying strength of the polar vortex, including sudden stratospheric warming (SSW) events. These results are in good agreement with previous studies and underpin the high utility of the entire RO record for investigating latitudinal, longitudinal, and temporal tropopause characteristics globally.

Color difference formulae evaluation by method of adjustment

2014 ◽  
Vol 11 (1) ◽  
pp. 89-94 ◽  
Author(s):  
A. Abbasi ◽  
M. Vik ◽  
M. Vikova
Keyword(s):  
Color Difference ◽  
Psychophysical Method ◽  
Psychophysical Experiment ◽  
Statistical Measures ◽  
Color Differences ◽  
The Mean ◽  
Different Color ◽  
Method Of Adjustment ◽  
Visual Results ◽  
Test Color

This experiment was carried out to test color difference formulae by method of adjustment. 6 different color centers (Red, Yellow, Green, Blue-Green, Blue and Violet) were chosen for this psychophysical experiment. 336 virtual sample pairs were prepared. The mean color difference was three CIELAB units. Each pair was assessed by a panel of 15 observers using psychophysical methods called Method of Adjustment. These visual data were used to test color-difference formulae: CIELAB, CMC, CIE94, DIN99d and CIE2000 with the help of simple statistical measures i.e., PF/3 and Stress. It was found that the visual results obtained from psychophysical method of adjustment show that CIE94 as well as DIN99d still perform well for small color differences.

Velocity and size characteristics of liquid-fuelled flames stabilized by a swirl burner

1990 ◽  
Vol 428 (1874) ◽  
pp. 129-155 ◽  
Keyword(s):  
Flow Rate ◽  
Elevated Temperatures ◽  
Outer Part ◽  
Arithmetic Mean ◽  
Recirculation Region ◽  
Reacting Flow ◽  
Air Velocity ◽  
Flow Rates ◽  
The Mean ◽  
Air Motion

Velocity and droplet size characteristics of an unconfined quarl burner, of 16 mm quarl inlet diameter, have been measured with a phase-Doppler anemometer at a swirl number of about 0.29: the Reynolds number of the flow was 30000, based on the cold bulk velocity of 30.4 m s -1 and the hydraulic diameter. The atomization was achieved by shear between the swirling air and six radial kerosene jets and the resulting Sauter and arithmetic mean diameters were about 70 and 50 μm respectively after injection: velocity characteristics are presented for three 5 μm-wide size classes, 10, 30 and 60 μm. The flows correspond to no combustion and combustion of natural gas with a heat release of 8 kW supplemented by liquid kerosene flow rates sufficient to generate 21.6 and 37.2 kW : the gas equivalence ratio was 0.45 and atomized kerosene at two flow rates increased the overall ratios to 1.64 and 2.53. In non­-reacting flow, droplets 30 μm and smaller are sufficiently small to be entrained by the mean air velocity towards the central part of the flow and into the swirl-induced recirculating air bubble. The 60 μm droplets are able to travel through the bubble uninfluenced by turbulent fluctuations in the air and are ‘centrifuged’ away from the centreline, through acquisition of a mean swirl velocity component, so that a large proportion of the kerosene volume flow rate lies at the edge of the swirling jet. Because larger droplets are centrifuged to the outer part of the flow, whereas the smaller are entrained towards the centreline, the Sauter and arithmetic mean diameters are, by 1.22 quarl exit diameters downstream of the quarl, approximately 65 and 36 μm at the outer part of the flow and 35 and 12 μm near the centreline in the inert flow. In reacting flow, droplets evaporate rapidly in regions of elevated temperatures and hence no droplets are found within the flame brush and recirculation region. The aerodynamic response of each size class to the air velocity is similar to inert flow so that the majority of the kerosene flow is centrifuged away from the flame. On exit from the quarl, the evaporation and burning rates cause the Sauter and arithmetic mean diameters to be about 70 and 50 μm and 60 and 30 μm at the inner and outer edges of the spray respectively. By 1.22 quarl exit-diameters from the exit of the quarl, the air motion entrains droplets smaller than about 30 μm towards the flame, at the inner edge of the spray, so that the Sauter and arithmetic mean diameters are 60 and 40 μm at the outer edge of the jet. There is comparatively little effect of changing the flow rate of kerosene because the combustion is controlled by the low available number of smaller droplets, although the Group combustion number corresponds to ‘cloud’ burning. The relative response of droplets to the mean and turbulent components of air motion, including the ‘centrifuging’ effect, can be scaled to other flows through dimensionless numbers defined in the text.

The effects of spatial variability of nitrous oxide emissions from grazed pastures on the sampling distribution of chamber measurements

2015 ◽  
Vol 154 (2) ◽  
pp. 223-241 ◽  
Author(s):  
D. L. GILTRAP ◽  
A. J. R. GODFREY
Keyword(s):  
Nitrous Oxide ◽  
Spatial Variability ◽  
Agricultural Soils ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Nitrous Oxide Emissions ◽  
Sample Mean ◽  
Grazed Pastures ◽  
Urine Patch ◽  
The Mean

SUMMARYChamber sampling is a common method for measuring nitrous oxide (N2O) emissions from agricultural soils. However, for grazed pastures, the patchy nature of urine deposition results in very high levels of spatial variability in N2O emissions. In the present study, the behaviour of the sample mean was examined by simulating a large number (9999) of random N2O chamber samples under different assumptions regarding the underlying N2O distribution. Using sample sizes of up to 100 chambers, the Central Limit Theorem did not apply. The distribution of the sample mean was always right-skewed with a standard deviation varying between 12·5 and 135% of the true mean. However, the arithmetic mean was an unbiased estimator and the mean of the sample mean distribution was close to the true mean of the simulated N2O distribution. The properties of the sample mean distribution (variance, skewness) were affected significantly by the assumed distribution of the emission factor, but not by distribution of the urine patch concentration. The geometric mean was also investigated as a potential alternative estimator. However, although its distribution had lower variance, it was also biased. Two methods for bias correcting the mean were investigated. These methods reduced the bias, but at the cost of increasing the variance. Neither of the bias-corrected estimators were consistently better than the arithmetic mean in terms of skewness and variance. To improve the estimation of N2O emissions from a grazed pasture using chambers, techniques need to be developed to identify urine patch and non-urine patch areas before sampling.

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