A bell-shaped curve will always have a peaked shape to demonstrate the location of the scores.
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A graph that depicts a normal probability distribution Show
What is a Bell Curve?A bell curve is the informal name of a graph that depicts a normal probability distribution. The term obtained its name due to the bell-shaped curve of the normal probability distribution graph. However, the term is not quite correct because the normal probability distribution is not the only probability distribution whose graph shows a bell-shaped curve. For example, the graphs of the Cauchy and logistic distributions also demonstrate a bell-shaped curve. Characteristics of a Bell CurveThe bell curve is perfectly symmetrical. It is concentrated around the peak and decreases on either side. In a bell curve, the peak represents the most probable event in the dataset while the other events are equally distributed around the peak. The peak of the curve corresponds to the mean of the dataset (note that the mean in a normal probability distribution also equals the median and the mode). The dispersion of the data on the bell curve is measured by the standard deviation. The probabilities of the bell curve and the standard deviation share a few important relationships, including:
The relationships described above are known as the 68-95-99.7 rule or the empirical rule. The empirical rule is primarily used to calculate the confidence interval of a normal probability distribution. The concept is extremely important in statistics due to the wide applications of the normal probability distribution. For instance, the normal probability distribution is used as a representation of the distribution of random variables whose real distribution is unknown. Bell Curve in FinanceIn finance, the normal probability distribution and the bell curve also play significant roles. Financial analysts frequently rely on the normal probability distribution in analyzing the returns of securities. The assumption of a normal distribution is fundamental in many pricing models that intend to predict future returns. However, one must be careful with the normal probability distribution assumption in finance. In reality, the returns on many securities tend to demonstrate a non-normal distribution. For instance, some distributions are skewed or with a kurtosis that is different from that of a normal distribution. Related ReadingsThank you for reading CFI’s guide to Bell Curve. To keep learning and advancing your career, the following CFI resources will be helpful:
Project forecasting is not a straightforward process – it’s a scientific and statistical exercise that deals with numerous interconnected variables. The value of these variables can have significant impact on your final prediction. Knowing the frequency distribution of these values enables you to make data-driven decisions. The type of frequency distribution is especially important when making predictions with the Monte Carlo simulation. In this article, we’ll explain the frequency distribution shapes that you will encounter most frequently. Normal DistributionThe normal distribution, also known as a Gaussian distribution or “bell curve” is the most common frequency distribution. This distribution is symmetrical, with most values falling towards the centre and long tails to the left and right. It is a continuous distribution, with no gaps between values. Normal distributions are found everywhere, for both natural and man-made phenomena. This could include time taken to complete a task, IQ test results, or the heights of a group of people. In project management, when performing estimations while you have no further information about the type of frequency distribution, it is usually best to assume a normal distribution. Skewed Distribution When a normal curve slopes to the left or right, it is known as a skewed distribution. The location of the long tail – not the peak – is what gives this frequency distribution shape its name. A long tail on the right is referred to as right-skewed or positively skewed, while a long tail on the left is referred to as left-skewed or negatively skewed. Positively skewed distributions are common in situations where there is a fixed lower boundary. For example, delivery of a component – if most deliveries happen within 3 days, the minimum value is 0, but the long tail could stretch far to the right if some deliveries are late. Negatively skewed distributions are less common in general, but still appear when fixed or near-fixed upper boundaries are in play. For example, a company that guarantees all orders will be delivered within 1 week will most likely see some faster deliveries, but most values clustering close to the 1 week point. One important fact about skewed distributions is that, unlike a bell curve, the mode, median and mean are not the same value. The long tail skews the mean and median in the direction of the tail. There is a very easy way to calculate the different average values using a histogram diagram. If you rely on average values to make quick predictions, pay attention to which average you use! Bimodal/Multimodal DistributionAll of the frequency distribution types that we’ve looked at so far have been unimodal – values cluster around a single peak. A bimodal distribution occurs when two unimodal distributions are in the group being measured. When more than two peaks occur, its known as a multimodal distribution. This distribution shape happens frequently when the measured data can be split into two or more groups. One example would be the throughput of all of your team’s tasks. If your team are using Classes of Service to tackle emergency tasks faster than regular tasks, you will most probably see a bimodal distribution. If you spot a bimodal frequency distribution, it’s worth checking if you can split the measured data into sub-groups to see the shape for each group. Uniform DistributionIn a uniform or rectangular distribution, every variable value between a maximum and minimum has the same chance of occurring. The probability of rolling a certain number on a dice or picking a certain card from the pack is described by this frequency distribution shape. This frequency distribution appears at the start of every project. A uniform distribution assumes that all samples from its population are equally probable. When rolling a die, all numbers on the die have an equal chance of coming up on each throw. Let’s say you have nineteen samples from a uniformly distributed population. In a uniform distribution there is a very high probability that the next sample will be between the min and the max of the previous samples. That means that you have a fairly good understanding of the range of your uniform distribution after having collected only twenty data points. Logarithmic/ParetoSome data sets have nearly all their frequency values clustered to one side of the graph. This frequency distribution shape is known as logarithmic. A common example of this in real life is found in distributions of wealth and income, with large numbers of people at the bottom but extreme outliers extending the tail to the right. This distribution type is often known as a Pareto distribution, named after famous Italian economist and sociologist Vilfredo Pareto. You’ve almost certainly heard of his 80-20 rule. For example, 80% of the wealth of a society is held by 20% of society, 80% of revenue comes from 20% of clients and 80% of productivity comes from 20% of your team. While the percentages are not always 80-20, this pattern appears mainly in financial estimation models. PERT/TriangularThe PERT and triangular frequency distribution types are both modelled from the same 3 values – a minimum, a maximum and a mode. This distribution type is especially useful when only a small amount of past performance data is available. It uses only three values as the inputs – a, m and b. While the triangular distribution is a simple shape made using straight lines between each of the 3 values, the PERT distribution assumes that the long tail values appear less frequently. The frequency distribution shape generated from these three values is then used to estimate likely completion times. Understanding the frequency distribution of your data is important for both input and output of your forecasts. Realistic outputs are simply impossible without accurate inputs. Calculations that rely on subjective estimates are risky – we recommend to always draw from your past performance data. What is the frequency distribution of your data? Have you used histogram diagrams to analyse it? Do you use histograms to make your estimations? Tell us about your experience in the comments! Do you find this article valuable? Rating: 4.7 stars (22 readers voted) Which measure of central tendency is most affected by extreme very high or very low scores?Answer and Explanation: Option A is the solution since the mean involves every point in the data set in its calculation, it becomes the measure of central tendency most susceptible to outliers or extreme values.
Which measure of central tendency occurs with the greatest frequency?There are three main measures of central tendency: the mode, the median and the mean. Each of these measures describes a different indication of the typical or central value in the distribution. What is the mode? The mode is the most commonly occurring value in a distribution.
Which measure of central tendency is most appropriate when it is desired to minimize the effect of a few extreme very high or very low scores?Mean is the most frequently used measure of central tendency and generally considered the best measure of it. However, there are some situations where either median or mode are preferred. Median is the preferred measure of central tendency when: There are a few extreme scores in the distribution of the data.
Which means of central tendency is most often referred to as the average?The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average.
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