Bell Curve and Normal Distribution Definition

Bell Curve and Normal Distribution Definition The term bell curve is used to describe the mathematical concept called normal distribution, sometimes referred to as Gaussian distribution. Bell curve refers to the shape that is created when a line is plotted using the data points for an item that meets the criteria of normal distribution. The center contains the greatest number of a value and, therefore, would be the highest point on the arc of the line. This point is referred to the mean, but in simple terms, it is the highest number of occurrences of an element (in statistical terms, the mode). Normal Distribution The important thing to note about a normal distribution is the curve is concentrated in the center and decreases on either side. This is significant in that the data has less of a tendency to produce unusually extreme values, called outliers, as compared to other distributions. Also, the bell curve signifies that the data is symmetrical. This means that you can create reasonable expectations as to the possibility that an outcome will lie within a range to the left or right of the center, once you have measured the amount of deviation contained in the data.This is measured in terms of standard deviations. A bell curve graph depends on two factors: the mean and the standard deviation. The mean identifies the position of the center and the standard deviation determines the height and width of the bell. For example, a large standard deviation creates a bell that is short and wide while a small standard deviation creates a tall and narrow curve. Bell Curve Probability and Standard Deviation To understand the probability factors of a normal distribution, you need to understand the following rules: The total area under the curve is equal to 1 (100 percent)About 68 percent of the area under the curve falls within one standard deviation.About 95 percent of the area under the curve falls within two standard deviations.About 99.7 percent of the area under the curve falls within three standard deviations. Item Nos. 2,3 and 4 are sometimes referred to as the empirical rule or the 68-95-99.7 rule. Once you determine that the data is normally distributed (bell curved) and calculate the mean and standard deviation, you can determine the probability that a single data point will fall within a given range of possibilities. Bell Curve Example A good example of a bell curve or normal distribution is the roll of two dice. The distribution is centered around the number seven and the probability decreases as you move away from the center. Here is the percent chance of the various outcomes when you roll two dice. Two: 2.78 percentThree: percentFour: 8.33 percentFive: 11.11 percentSix: 13.89 percentSeven: 16.67 percentEight: 13.89 percentNine: 11.11 percentTen: 8.33 percentEleven: 5.56 percentTwelve: 2.78 percent Normal distributions have many convenient properties, so in many cases, especially in physics and astronomy, random variations with unknown distributions are often assumed to be normal to allow for probability calculations. Although this can be a dangerous assumption, it is often a good approximation due to a surprising result known as the central limit theorem. This theorem states that the mean of any set of variants with any distribution having a finite mean and variance tends to the normal distribution. Many common attributes such as test scores or height follow roughly normal distributions, with few members at the high and low ends and many in the middle. When You Shouldn't Use the Bell Curve There are some types of data that dont follow a normal distribution pattern. These data sets shouldnt be forced to try to fit a bell curve. A classic example would be student grades, which often have two modes. Other types of data that dont follow the curve include income, population growth, and mechanical failures.

Consider whether the different tests for certainty of objects applicable to fixed trusts and discretionary trusts are appropriate Essays

Consider whether the different tests for certainty of objects applicable to fixed trusts and discretionary trusts are appropriate Essays Consider whether the different tests for certainty of objects applicable to fixed trusts and discretionary trusts are appropriate Essay Consider whether the different tests for certainty of objects applicable to fixed trusts and discretionary trusts are appropriate Essay Essay Topic: Law The Certainty of object form one of the three requirements which must be satisfied to validate a trust. The fundamental principle is that to properly enforce a trust it must have cestque tui trust and it must be possible to establish who the beneficiaries are1. These apply to both fixed and discretionary trusts, which convey the expressed wish of a testator. In effect it is incumbent on the settlor to enable some means of ascertaining the intended beneficiary; and appropriate tests for objects would be needed to ensure the trust is properly enforced. Traditionally a general rule applied to all trusts; the trustee has a duty to administer the trust according to the trust instrument and so would need to know exactly how many beneficiaries there are, thus must draw up a fixed list2. Under a fixed trust the testator would express the beneficiary to whom the trust was intended and therefore the object is often clear. However where the beneficiaries are of a wide class conceptual uncertainties commonly arise and it would therefore require interpretation. Such a situation arose in Broadway3 the trust was void for uncertainty as the whole range of objects could not be ascertained. It is generally accepted that the terms in a fixed trust are precise enough to comprise a complete list test. However where the testator aims to give to the benefit of a large number of people a discretionary trust is most useful. This is because no individual potential beneficiary has an interest on the fund until the trustees discretion is exercise. More recently, the complete list has proved especially problematic for the increasingly popular large corporate trusts, which tends to distribute amongst a very wide class- (by applying Broadway), these would frequently fail for uncertainty. 4 One the one hand, because the court is obliged to enforce the trust5, the use of a complete list test is essential to manage a trust. When applied to fixed trusts, it reflects the testators determination to ensure the trust is executed exactly as he intended. Thus if the executor was uncertain, the income would belong to the settlor on resulting trust. In such circumstances it seems plausible that whilst reforms in McPhail6 only changed the law in relation to discretionary trusts, Broadway continues to regulate fixed trust. Mcphail 7drew upon similarities between powers and discretionary trusts which Broadway overlooked. Firstly although trustees for discretionary trusts have an imperative duty to execute the fund, like a power, they are given the choice of how this should be done and so proposed to assimilate the validity test for trusts with that which applies to powers. Overall a complete list was deemed too rigid and instead ReGulbekian8 should also apply to discretionary trusts. Whilst it was the ideal test for mere powers, its application to discretionary trust would prove objectionable. 9 In addition to the need for conceptual certainty, there was also need for sufficient practical certainty in its definition to be carried out. Therefore even if a class is conceptually certain it could still be invalid if administratively unworkable. However to uphold the principle in Broadway would be to order an equal distribution in which every beneficiary share. This would probably defeat the settlors intention; as equal division among all may produce a result beneficial to none. 10 Overall it considered whether the is or is not test was a semantic or evidential one; a question which, if unresolved, could lead to an irrational development of law. The issue was addressed in ReBaden, however there were three distinct reasoning; At one end of the spectrum Stamp LJ, imposed the most rigorous test, question whether he is, or is not, a member of a conceptually certain class. However whilst accepting that it would be impossible to devise a complete list, he emphasised a need to obtain the widest possible range of objects. In practice, the difference between this test, and the `complete list test, is very slight. Therefore while it seems keenest to consider the maximum number of beneficiaries, the approach makes it most vulnerable to failing of conceptual uncertainty. Megaw LJ took an almost opposing view, which was also the softest approach. Identifying a substantial number of people, within the terms set out by the settlor. Whilst it classifies when a trust would be valid, it does not guide the trustee on how to measure uncertainty in the boundaries of the class. This inability to distinguish between conceptual certainty and evidential certainty therefore makes it impractical. Sach LJ was a middle ground approach. 11 The trust would succeed if it would be possible to determine in theory whether any given person was inside or outside of the class. Where objects are less like a class and appear rather as applicants to a fund for which they might qualify for a distribution, (whether they actually receive funds lies at the discretion of the trustee whose only obligation is to distribute), therefore the trustee could justify their act based on a solid test of whether or not any individual distribution is legitimate. 12 The Courts generally adopt the Sach approach, largely because it is least likely to fail for administrative unworkability13. It only imposed the need for conceptually certainty, thus evidential difficulties would not affect the validity of a trust. However problems with administering the trust itself could still exist for instance where the words in the trust are clear, but the ambit is so wide that the costs of ascertaining the members would outweigh the value of fund. However precedent suggests this is unlikely. 14Another problem is that he focused on the similarities between a trust and a power, without addressing the differences. Clearly the duty under a trust is more onerous and the consequences for negligence are higher than for a power who can act free from regulation. However on a positive note the is or is not test, does not oblige trustees to consider all the potential candidates, so it may be easier to prove that their actions were for the benefits of the trust. Whilst Sachs approach may enable the trustee to provide a theoretical justification, it does not ascertain every object. Certainty of objects also apply to testamentary gifts subject to condition precedent. Ambiguity as to whom the testator intended to benefit, would give rise to the same problems which affect trusts. In Re Barlow, the question arose as to the meaning of friends of mine and the courts contemplated which test should apply. Although bearing similarities to Megaw, the final ruling did not fully adopted any of the approaches in Re Baden, inevitably it is questionable whether the is or is not is appropriate. Despite the difficulties in applying suitable tests the courts is clearly more inclined to give effect to a trust than to invalidate one. This was demonstrated in the lack of unanimity in ReTuck which deferred the Chief Rabbi to provide definition of Jewish women should difficulties arise, although there was no consensus in the judges the trust was still held valid.