Category: Maths Notes PPT

Division results in a different number as a result of dividing a number by another number. Dividends are numbers or integers that get divided, and the integer that divides a given number is the divisor. The remainder of a divisor is that number that does not divide the number entirely. Division symbols are represented by a ÷  or a /. Therefore, we can represent the division method as follows;

Dividend = Quotient × Divisor + Remainder

If the remainder is zero, then; Dividend = Quotient × Divisor

Therefore, Quotient = Dividend ÷ Divisor

Definition of a Partial Quotient

An approach to solving large division problems by using partial quotients is called a partial fraction. By taking a more logical approach to the problem, the student is able to see it less abstractly.

If you want to try this technique in your classroom, you might want to start with the Box Model/Area Model. Partial Quotients and the Box Method are similar in approach, but the Box Method is structured differently and is a good introduction to the quantitative method.

Using Partial Quotients as a Division Strategy

In the division of repeated subtraction and partial quotients, there is a strong correlation. This correlation makes dividing long divisions easy to comprehend and apply. You can easily divide partial quotients according to the three simple steps outlined here. The final quotient can be calculated by adding up each multiplier in the repeated subtraction division method. Long division of numbers can be easily performed by dividing using partial quotients. A fraction of a second after the partial quotient division of the numbers has been performed, the quotient and remainder will be displayed.

How to Divide Using the Partial Quotients Method?

To solve basic division problems, the partial quotients approach (also known as chunking) employs repetitive subtraction. When dividing a big number by a small number.

• Step 1– Find out an easy multiple of the divisor and deduct from the dividend (for example 100 ×, 10 ×, 5 × 2 ×, etc.) 

• Step 2 – Continue subtracting until the large number is reduced to zero or the remainder equals the divisor.

• Step 3 – To find the division answer, add up the multipliers of the divisor that were used in the repeated subtraction.

The partial quotient approach is shown in the diagram below. More explanations and solutions can be found further down the list

Using Partial Quotients to Divide One-Digit Numbers

Method of Partial Division

The use of partial quotients is useful for dividing large numbers. Using partial quotients you can divide the problem into smaller pieces and simplify division. All of the bits are then added back together to get the total.

 Let’s try it with 654 ÷ 3.

Step 1 – Start by subtracting multiples of 3 until you reach 0.

A multiple of 3 that goes into 654 is 600, because 3×200=600. Subtract 600 from 654.

You have 54 left. Now subtract another multiple of 3. You can use 30, since 3×10=30.

You have 24 left. Keep going! Subtract 24, since 3×8=24.

You’ve reached 0, so move to step 2.

Step 2 – Now, see how many times of ‘3’ it took to reach 654.

You broke 654 into 600, 30, and 24. Add the number of times it took 3 to reach each of those numbers.

200 + 10 + 8 = 218

So, 654 ÷ 3 = 218!

Divide Decimals with Partial Quotients 

Using Partial Quotient, You can divide Decimals with Remainders. This will help you find the problem. Make your answer as precise as possible.

Partial Quotients for Dividing by 2-Digit Numbers

Using partial quotients is also an option for dividing large numbers.

Let’s try it with 5,520 ÷ 23.

Step 1 – Start by subtracting multiples of 23 until you reach 0.

A multiple of 23 that goes into 5,520 is 4,600, because 23 × 200 = 4,600. Subtract 4,600 from 5,520.

You have 920 left. Now subtract another multiple of 23. You can use 690, since 23 × 30 = 690.

You have 230 left. Subtract 230, since 23 × 10 = 230.

You’ve reached 0, so move to step 2.

Step 2 – Now, see how many times 23 went into 5,520.

You broke 5,520 into 4,600, 690, and 230. Add the number of times it took 23  to reach each of these –

200 + 30 + 10 = 240

So, 5,520 ÷ 23 = 240! 

Have you ever thought about what length of string you need to wrap a gift or how many meters you walked around a square park or how many marbles you need to for the boundary of your room?

Perimeter finds its application in various fields and in our day to day life. Suppose uncle John completes two rounds of a square park every morning. Now, to find out the perimeter of the square park to calculate how many kilometers uncle John covers every day. Similarly, a farmer needs to know the cost of fencing his field in order to save his crops from ruminant animals. To find the cost of fencing he needs to know the perimeter of his field.
                       

What is the Perimeter of a Square?

Any shape that can be laid on a flat surface is called a two-dimensional object. Square is one of the 2D-Shape having four equal sides and four corners of 90 degrees each. The length of the boundary or sides of any two-dimensional shape is called the perimeter. The perimeter of a Square is the sum of the length of all the four sides of a square.

 

The Perimeter of Square Formula:

The perimeter of a square = Sum of all the Sides     (or)

                                                = Side+Side+Side+Side.

 

Now, as we know, the length of all the four sides of a square are equal so the perimeter is four times the length of any side of a square.

The perimeter of a square = 4 x sides.

Let’s take an example of a square having side ‘a’.

Perimeter  = side x side x side x side

                    = a x a x a x a

                    = 4 x a

                    = 4a

 

Solved Examples

Below are the Perimeter of Square Formula examples:

Problem 1:

Preeti takes three rounds of a square park of side 70m. Find the distance traveled by Shaina.

Solution: Side of the square park is 70 meter.

We know, the perimeter of a square = sum of all the sides

                                   = 70 x 70 x 70 x 70 meters

                                   = 4 x 70 meters

                                   = 280 meters.

Therefore, the boundary of the square park measures 280 meters.

If Preeti takes three rounds of the park, then the total distance traveled by Preeti

= 3 x 280 meters

= 840 meters.

 

Problem 2: 

The lid of a square box of side 50cm is sealed with tape. What is the length of the tape?

Solution: 

Length of the side of the square box is 50cm. 

The length of the tape is equal to the perimeter of the box

The perimeter of the box = 4 x side

                                              = 4 x 50 cm

                                              = 200 cm.

Thus, the length of the tape is 200cm.

 

Problem 3:

If the perimeter of a square is 20m, then what is the side of the square?

Solution:

We know, 

Perimeter = 4 x side

Side = Perimeter / 4

        = 20 / 4 cm

        = 5 cm.

 

Problem 4: 

What is the Perimeter of a square and also find the cost of fencing a square park of 250 meters at the rate of Rs 20 per meter.

Solution: To find the cost of fencing a square park we need to find out the length of the boundary of a square park.

The perimeter of a square = 4 x sides

 If the side of the square park is 250 meters then,

 The perimeter of a square = 4 x 250

                                                 = 1000 meters.

The cost of fencing 1 meter is Rs 20 

So, the cost of fencing 1000 meters will be 1000×20  i.e,  20,000.

Therefore the cost of fencing a square park of sides 250 meters is Rs 20,000.

Finding the Perimeter of the Square using its Area 

Suppose the question does not specify the side length of the square. Instead, it gives you the area of the square and asks you to find the perimeter. 

Then, you can find the perimeter of the square by using the formula of the area of a square. Follow the steps mentioned below to understand how to determine the perimeter,

1. Read the question carefully and note down the area of the square. 

2. You can use the area to find the side length of the square. Here’s how: 

Area of square (A)= Side × Side 

So, Side = √A

3. Once you have obtained the side length, you can put its value in the formula of the perimeter of the square. 

Perimeter = 4 × √area 

Finding the Area of Perimeter using its Diagonals 

Like the above question, some questions will only specify the diagonal length of the square instead of the side length. In such cases, you can follow the steps mentioned below to find the perimeter of the square.

1. Go through the question thoroughly and note down the diagonal length. 

2. Use the diagonal length to find the side length of the square using the formula: 

Side = diagonal/√2 

3. Then apply the length of the side, obtained in the above step, in the formula of the perimeter of the square. You will get, 

Perimeter = 4 × diagonal/√2 

This way, you can find the perimeter of the square using only the length of the diagonals. By learning these formulas, you will be able to solve any question, based on the perimeter of a square, asked in your math exam.

A pint is a unit of volume or capacity in both the imperial and United States customary measurement systems. The symbol ‘pt’ is used to represent a pint. In both systems, a pint is traditionally about ⅛ of a gallon. The British Pint is about ⅕ larger than American Pint as both the systems are defined differently.

In the British system, the units for the dry and liquid measure are similar, the single British pint is equivalent to 34.68 cubic inches (568.26 cubic cm) or one- eight gallon. In the United States, the unit for dry systems is slightly different from liquid measures. A U.S. dry pint is 33.46 cubic inches (550.6 cubic cm), while a liquid pint is 20.9 cubic inches (473.2 cubic cm). In each system, two cups (unit of volume in the British Imperial and United States Customary systems of measurement) makes a pint, and two pints equals a quart (a unit of capacity in the United States Customary systems of measurement and the British Imperial).

What is an Example of 1 Pint?

A pint is equivalent to 2 cups (for example, a large glass of milk).

uploaded soon)

The value of 1 Pint = 2 Cups = 16 Fluid Ounces.

A unit quart (qt) is used in place of a pint for measuring many cups of liquid together.

uploaded soon)

The value of 1 quart (qt) is similar to 4 cups or 2 pints.

The Value of 1 Quart = 2 Pint = 4 Cups = 32 Fluid Ounces

If we still need to measure more liquid, then we can use the unit gallon in place of quat(qt).

1 Gallon =  8 Pints = 16 Cups = 4 Quarts

uploaded soon)

Gallon is the largest measurement of liquid.

Note: A Quart is a Quarter of a Gallon.

Pint of Water

The value of 1 US fluid pint of water weighs about a pound (16 ounces). This gives rise to a renowned statement, A pints, a pound, the world around. The measure of US pint of water is approximately around 1.04318 pounds, and this statement does not hold throughout the world.

It is due to the imperial pint which was also the standard measure in New Zealand, Australia, Malaya, India, and other British colonies weighs 1.2528 pounds, which gives rise to a popular statement for the imperial pint “ a pint of water that is pure weighs a quarter and a pound).

What is Half Pint?

Half-pint or half a pint is equivalent to 8 fluid ounces (1 cup) or 16 tablespoons (0.2 litres).

A 375 ml of pint in the US and the Canadian maritime provinces is sometimes referred to as a pint, and a  200 ml bottle is known as half a pint, looking back to the days when liquor came in US pints, fifths, quartz,  and gallon.

A standard 250 ml of beer in France is known as un demi (” a half”), originally meaning half a pint.

Imperial Pint

The value of an imperial pint is equivalent to one – eight imperial gallons.

The Value of 1 Imperial Pint equals to

= ⅛ Imperial Gallon

= ½ Imperial Quart

= 4 Imperial Gills

= 20 Imperial Fluid Ounces

= 568.26125 millilitres exactly

≈ 34.677429099 cubic inches

≈ 1.0320567435 US dry pint

≈ 1.2009499255 US liquid pint

≈ 19.215198881 US fluid ounces

≈ The volume of 20 oz (567 g) of water at 62.F (16.7 C).

US Liquid Pint

1 US Liquid Pint is equals to

= ⅛ US liquid Gallon

= ½ US liquid quart

= 2 US cups

= 4 US fluid gills

= 16 US fluid ounces

= 128 US fluid drams

= 28.875 cubic inches (exactly)

= 473.176473 milliliters ( exactly)

≈ 0.83267418463 imperial pints

≈ 0.85936700738 US dry pints

≈16.65348369 imperial fluid ounces

≈The volume of 1.041 lb (472 g) of water at 62° F ( 16.7° C)

US Dry Pint

In the US, the dry pint is equal to a sixty-fourth of a bushel

1 US dry Pint is equals to

= 0.015625 US bushels 

= 0.125 US Pack

= 0.5 US Dry gallon

= 33.6003125 US dry quarts

≈ 550.6104713575

≈ 0.96893897192092

≈ 1.1636471861472 US pints

Facts to Remember

• One pint is equivalent to one-eighth of a gallon and half of a quart.

• One Pint is equal to 2 cups or 16 ounces.

• 2 Pint is equal to 1 Quart.

• 8 Pint is equal to 1 gallon.

Examples:

1. Convert 20 pt (US) to cups (US).

Solution:

The value of 1 pt (US) = 2 cup (US)

The value of 1 cup (US) = 0.5 PT (US)

Accordingly,

20 pt (US) = 20 × 2 cup (US) = 40 cup (US).

2. Convert 20 pt (US) to Gallons (US).

Solution:

The value of 1 pt (US) = 0.125 gal (US)

The value of 1gal (US) = 8 pt (US)

Accordingly,

20 pt (US) = 20 0.125 gal (US) = 1.875 gal (US).

Polyhedron definition states that “a three-dimensional structure in Euclidean geometry, made up of a finite number of polygonal faces”.

The boundary between the interior and the exterior of a solid is a polyhedron.

Polyhedrons, in general, are named according to the number of faces.

Parts of Polyhedron

The Polyhedron has three parts namely:

The face is a flat surface that makes up a polyhedron which is regular polygons.

Edge is the region where the two flat surfaces meet to form a line segment.

Vertex, also known as a corner, is a point of intersection of the edges of the polyhedron.

uploaded soon)

Types of Polyhedron

There are three main types of Polyhedron:

1. Prisms

A prism is a polyhedron consisting of an n-sided polygonal base, a second base that is a translated copy of the first, and no other faces that connect the two bases to corresponding sides.

Prisms are named according to their cross-sections(polyhedron faces).

(Image will be uploaded soon)

uploaded soon)

uploaded soon)

uploaded soon)

2. Platonic Solids

A regular, convex polyhedron is a Platonic solid in three-dimensional space. It is constructed of congruent, regular, polygonal faces that meet at each vertex with the same number of faces.

Platonic solids are of five types based on Polyhedron faces and polyhedron shapes:

• Tetrahedron

• It has 4 faces, 4 Vertices, and 6 Edges.

• It has 3 triangles that meet at each vertex.

• The surface area of tetrahedron= [sqrt{3} a^{2}]

• The volume of tetrahedron= [frac{sqrt{2}}{12} a^{3}]

uploaded soon)

• Cube

• It has 6 faces, 8 Vertices, 12 Edges.

• It has three squares that meet at each vertex.

• The surface area of the cube= 6a2

• The volume of the cube = a3

uploaded soon)

• Octahedron

• It has 8 faces, 6 Vertices, 12 Edges.

• It has 4 triangles that meet at each vertex.

• The surface area of Octahedron = [2 sqrt{3} a^{2}]

• The volume of Octahedron = [frac{sqrt{2}}{3} a^{3}]

uploaded soon)

• Dodecahedron

• It has 12 faces, 20 Vertices, 30 Edges.

• It has 3 pentagons that meet at each vertex.

• The surface area of Dodecahedron= [3 sqrt{25 + 10sqrt{5}} a^{2}]

• The volume of Dodecahedron = [frac{15 + 7sqrt{5}}{4} a^{3}]

uploaded soon)

• Icosahedron

• It has 20 faces, 12 Vertices, 30 Edges.

• It has 5 triangles that meet at each vertex.

• The surface area of Icosahedron = [5 sqrt{3} a^{2}]

• The volume of Icosahedron = [frac{5(3 + sqrt{5})}{12} a^{3}]

uploaded soon)

3. Pyramids

A pyramid is a polyhedron created by connecting a polygonal base and a point, called the apex. A triangle, called a lateral face, is formed by any base edge and apex. It is a conical solid with a foundation of polygons.

The types of pyramids are named after the base of a pyramid.

uploaded soon)

uploaded soon)

uploaded soon)

uploaded soon)

uploaded soon)

Counting Polyhedron Faces, Edges, and Vertices

Euler’s formula relates the number of faces, vertices, and edges of any polyhedron. This formula is used in Counting Polyhedron Faces, Edges, and Vertices.

Euler’s formula is given as follows: 

F + V – E = 2

Where F = Number of Faces

V = Number of Vertices

E = Number of Edges

Problems on Polyhedron Faces, Edges, and Vertices

1) The Polyhedron has 6 faces and 12 edges. Find the number of Vertices. Also, name the type of Polyhedron.

Ans: Here to find the number of vertices we will use Euler’s formula,

F+ V – E = 2

From the question F = 6, E = 12, V = ?. Substituting these values in the Euler’s formula we get,

6 + V – 12 = 2

V – 6 = 2

V = 8

Here we can conclude that the Polyhedron is a Cube.

2) The Polyhedron has 5 faces and 6 vertices. Find the number of edges. Also, name the type of Polyhedron.

Ans: Here we will use Euler’s formula to find the number of edges,

F + V – E = 2

From the given data F = 5, V = 6, E = ?. Substituting these values in the Euler’s formula we get,

5 + 6 – E = 2

11 – E = 2

E = 9

Therefore the polyhedron is a Triangular Prism.

Fun Facts

• The word Polyhedron comes from the Greek words “poly” meaning many, and “hedron,” meaning surface.

• Polyhedron means that to form a 3-dimensional shape, numerous flat surfaces are connected.

• < p role="presentation">Polyhedra is the plural of a polyhedron.

The largest number that divides each of the supplied numbers without leaving any remainder is the highest common factor (HCF) of two or more given numbers. The lowest of the common multiples of two or more numbers is called the least common multiple (LCM). Learning HCF and LCM in mathematics is important since it helps us in solving day-to-day problems involving grouping and sharing.

For example, the HCF of 24 and 36 is 12, as 12 is the greatest number that can divide both the numbers without leaving any remainder. In the same way, the LCM of two or more numbers is the lowest number that is a common multiple of the given numbers.

What are the Methods to Find HCF and LCM?

There are a number of methods to find the HCF and LCM of given numbers. But most common methods are:

Using Prime Factorization Methods to Find HCF and LCM

A factor is a number that divides a number exactly into a different number. For example, 5 divides 35 into 7, hence 5 is a factor of 35, while 7 is also a factor of 35. One is a factor of every number. The number itself is also a factor of any number.

Factorization is the method of writing a number as a product of several of its factors. Factorization is not considered meaningful as compared to division, but it finds its use when we wish to find the simplest constituents of a number and to represent a number as a product of the same.

Example 1: Find the HCF of 50 and 75 by the prime factorization method.

Ans: Prime factors of 50 = 2 × 5 × 5

Prime factors of 75 = 3 × 5 × 5

If we analyse factors of both numbers, the common factors are 5 × 5.

Therefore, the HCF of (50, 75) is 25.

Example 2: Find HCF of 390, 702, and 468 by using the prime factorization method.

Ans: Prime factors of 390 = 1 × 2 × 3× 5 × 13. 

Prime factors of 702 = 1 × 2 × 3 × 3 × 3 × 13. 

Prime factors of 468 = 1 × 2 × 2 × 3 × 3 × 13. 

Here, the common factors of 390, 702, and 468 are 1, 2, 3, 13. 

Therefore, HCF of 390, 702, and 468 is 2 × 3 × 13 = 78.

LCM by Prime Factorization

Prime factorization is one of the methods to find the LCM of two or more numbers. We can similarly separately find the prime factors of any numbers that are given and then find the LCM by identifying the common factors among them.

To calculate the least common multiple of given numbers, we follow the steps given below:

List all the prime factors of the given numbers and note down the common prime factors.

The LCM of the numbers is the calculated by-product of the common prime factors as well as the uncommon prime factors of the numbers.

Example 1: What is the L.C.M of 36 and 14 by using the prime factorization method?

Ans: To find the LCM, we have to multiply all prime factors. But note that the common factors are included only once.

The prime factors of 36 = 2 × 2 × 3 × 3 = 2² × 3².

The prime factors of 14 = 2 × 7.

L.C.M = 2² × 3² × 7 = 2 × 2 × 3 × 3 × 7 = 252.

Therefore, the least common multiple (L.C.M) of 36 and 14 = 252.

Example 2: What is the L.C.M of 5, 4, and 16 by using the prime factorization method?

Ans: To find the LCM, we have to multiply all prime factors. But note that the common factors are included only once.

The prime factors 5 = 5 × 1.

The prime factors 4 = 2 × 2.

The prime factors 16 = 2 × 2 × 2 × 2 = 2⁴.

L.C.M = 2⁴ × 5 = 2 × 2 × 2 × 2 × 5 = 80.

Therefore, the least common multiple (L.C.M) of 5, 4, and 16 = 80.

Conclusion

The highest common factor between two or more numbers is known as the HCF. The “Greatest Common Divisor” is another name for it. Similarly, the lowest number that is a common multiple of the given numbers is the least common multiple (LCM) of those numbers.

Definition with Example

Probability deals with predicting the likelihood of an event. Many events cannot be predicted with total certainty, so the best we can say is how likely they are to happen, using the idea of probability. Probability is primarily a branch of mathematics, which studies the consequences of mathematical definitions and real-life entities. The probability of an event is expressed as a number 0 and 1, 0 indicates the impossibility and 1 indicates the certainty of an event. The higher the probability shows that the more likely it is that the event will occur. A simple example is the tossing of an unbiased coin. Since the coin is unbiased, there are two probable outcomes, either its heads or tails; the probability of “heads” is equal to the probability of “tails”; there are no other outcomes that are possible, assuming the coin lands flat. So the probability of either “heads” or “tails” is ½ or 0.5 or 50%. An event having the probability of 0.5 is considered to have equal odds of occurring and no occurring. The probability that the coin will land without either side facing up is 0 because either “heads” or “tails” must be facing up.

Calculating probabilities in a situation like a coin toss is upfront because the outcome of the coin lands without either of the sides facing up is 0. Each coin toss is an independent event; the outcome of one trial does not affect the following ones. No matter how many times one side lands facing up, the probability that it will do so at the next toss is always 0.5 (50%). The mistaken idea that several consecutive results (seven “heads” for example) make it more likely that the next toss will result in a “tails” is known as the gambler’s fallacy, one that has led to the downfall of many a bettor’s.

Probability theory had its start in the 17th century, when two French mathematicians, Blaise Pascal and Pierre de Fermat carried on a correspondence discussing mathematical problems dealing with the games of chance. The modern applications of probability theory run on the extent of human inquiry and include aspects of computer programming, astrophysics, music, weather prediction, risk management, market assessment, entitlement analysis, environmental regulation, and financial regulation and medicine.

To measure probabilities, mathematicians devised the following formula to find the probability of an event: 

Probability of an Event Happening

P(A)=Total Number of Ways Event”A” Can OccurTotal Number of Possible Outcomes

Or in another way in its simplest form, probability can be expressed as the total number of occurrences of a targeted event divided by the total number of occurrences plus the total number of failures (this adds up to the total of possible outcomes):

P(A) = P(a)/P(a)+P(b)

Statistics

Statistics is a form of mathematical analysis for a given set of data or real-life studies that uses quantified models, representations and synopses to reach the results. Statistics studies methodologies to gather, review, analyze and draw conclusions from the experimental dataset. Some statistical measures include several, mode, median, regression analysis, skewness, kurtosis, variance, and analysis of variance.

Probability and Statistics

Probability is the probability of anything happening — how likely an occurrence is to occur. The study of data, including how to collect, summarise, and present information, is known as statistics. Probability and statistics are two academic subjects that are related but not identical. Probability distributions are frequently used in statistical analysis, and the two disciplines are frequently studied together.

Relational Symbols

Mathematical relations are represented by relational symbols, which express a connection between two or more mathematical objects or concepts.

Understanding Statistics

Statistics is a term used to summarize a process that is used to characterize a data set. If the data set depends on a sample of a larger population, then one can develop interpretations about the population primarily based on the statistical outcome from the samples. Statistical analysis involves the process of gathering, reviewing, evaluating data and then summarizing the data into a mathematical form or statistical outcome.

More generally statistical methods are used to analyze large volumes of data and their properties.

Statistics is used in various disciplines such as psychology, business, social sciences, humanities, government, medical and manufacturing. Statistical data is gathered using a sample procedure. There are two types of statistical methods that are used in analyzing data: descriptive statistics and inferential statistics. Descriptive statistics are used to summarize data from a sample exercising the mean or standard deviation. Inferential statistics are used when data is considered as a subclass of a specific population.

Types of Statistics

Statistics is a general, broad term, so it is natural that inside that umbrella there exist a number of different models.

Mean: A mean is the mathematical average of a group of two or more numbers. The mean for a specified set of numbers can be computed in multiple ways, including the arithmetic mean, which shows how well a specific commodity performs over time, and the geometric mean, which shows the performance results of an investor’s share invested in that same commodity over the same period.

Regression Analysis: Regression analysis determines the point to which specific factors such as interest rates, the price of a product or services, or particular industries influence the price variations of an asset. This is portrayed in the form of a straight line called a linear regression line.

Skewness: The degree of a set of experimental data in which the data varies from the standard distribution is known as skewness. In the case of most of the data sets, like stock prices and commodity returns, the data sets have either positive skew, a curve slanted toward the left of the data average, or negative skew, a curve slanted toward the right of the data average.

Kurtosis: Kurtosis measures whether the experimental data is light-tailed (less outlier-prone) or heavy-tailed (more outlier-prone) than the normal distribution. Data sets with high kurtosis have heavy tails, or outliers, which implies greater investment risk in the form of occasional wild returns. Data sets with low kurtosis have light tails, or lack of outliers, which implies lesser investment risk.

Variance: The measurement of the span between the numbers in a data set is called Variance. The variance measures the distance of every number in the data set through its mean. Variance can help to determine the risk an investor might accept when buying an investment plan.