Chapter 8 The Normal Distribution - The Normal Distribution

Oct 3, 2026

Uploaded by Kurz

Chapter 8 The Normal Distribution introduces how and why the normal distribution models real-world measurements. It highlights using statistical tables or graphic calculators to solve problems and stresses that solution accuracy depends on the tools used. The chapter motivates the topic through a survey of adult heights and an activity for collecting about 100 data points. It then examines frog-length data, explaining the bell-shaped pattern, the relationship between mean and median/mode, and the proportion within one standard deviation of the mean.

LanguageEnglish

CategoryExam

FormatPDF

Pages22

Reading Time55 min

AccessFree to download

Objectives

8.0 Introduction

Activity 1 Data collection

8.1 Looking at your data

Chapter 8 The Normal Distribution 8 THE NORMAL DISTRIBUTION Objectives After studying this chapter you should • appreciate the wide variety of circumstances in which the normal distribution can be used; • be able to use tables of the normal distribution to solve problems; • be able to use the normal distribution as an approximation to other distributions in appropriate circumstances. Statistical tables are available in many books and can also be found online. You should note that the accuracy of your solutions will depend on the tables (or graphic calculator) you are using. 8.0 Introduction The tallest accurately recorded human being was Robert Wadlow in the USA. On his death at the age of 22 he was 272 cm (8 feet 11.1 inches) tall. If you were an architect and you had to design doorways in a building you would clearly not make them all 9 feet high-most ceilings are lower than this! What height should the ceilings be? In 1980 the Government commissioned a survey, carried out on 10 000 adults in Great Britain. They found that the average height was 167.3 cm with SD (standard deviation) 9.1 cm. You cannot make a door size that everyone can fit through but what height of door would 95% of people get through without stooping? This chapter should help you find the answer. Activity 1 Data collection There are many sets of data you could collect from people in your group, such as heights, weights, length of time breath can be held, etc. However, you will need about 100 results to do this activity properly so here are a few suggestions where large quantities of data can be collected quickly. 1. Lengths of leaves Evergreen bushes such as laurel are useful-though make sure all the leaves are from the same year's growth. 151 Scan: free demo class MEGA LECTURE · Stuck on this topic? Free 1-on-1 demo class Take a free test WhatsApp a tutor +92 336 7801123 Chapter 8 The Normal Distribution 2. Weights of crisp packets Borrow a box of crisps from a canteen and weigh each packet accurately on a balance such as any Science laboratory would have. 3. Pieces of string Look at 10 cm on a ruler and then take a ball of string and try to cut 100 lengths of 10 cm by guessing. Measure the lengths of all the pieces in mm. 4. Weights of apples If anyone has apple trees in their garden they are bound to have large quantities in the autumn. 5. Size of pebbles on a beach Geographers often look at these to study the movement of beaches. Use a pair of calipers then measure on a ruler. 6. Game of bowls Make a line with a piece of rope on the grass about 20 metres away. Let everyone have several goes at trying to land a tennis ball on the line. Measure how far each ball is from the line. Try at least two of these activities. You will need about 100 results in all. To look at the data it would help to have a data handling package on a computer. 8.1 Looking at your data The data shown on the opposite page gives the length from top to tail (in millimetres) of a large group of frogs. This has been run through a computer package so you can see some useful facts about the data. In the computer analysis you will see that most of the frogs are close to the mean value, with fewer at the extremes. This 'bellshaped' pattern of distribution is typical of data which follows a normal distribution. To obtain a perfectly shaped and symmetrical distribution you would need to measure thousands of frogs. Does your data follow a 'bell shaped' pattern? You may notice that median ≈ mean ≈ mode, as might be expected for a symmetrical distribution. From the analysis of data you also see that the mean is 90.9 mm and the standard deviation is 11.7 mm. Now look at how much of the data is close to the mean, i. e. within one standard deviation of it. From the stem and leaf table you can see that 74 frogs have a length within one standard deviation above the mean and 59 within a SD below th

FAQs

What are the main objectives of Chapter 8?

Students learn where the normal distribution can be used, how to use normal distribution tables to solve problems, and how to use the normal distribution as an approximation when appropriate.

How can statistical tables be used to solve normal distribution problems?

The chapter notes that tables (or a graphic calculator) are used to obtain probabilities or proportions for values under the normal model, and accuracy depends on the particular tables/calculator used.

What does the frog data suggest about the normal distribution?

Most frog lengths cluster near the mean, forming a bell-shaped pattern. The analysis also shows median ≈ mean ≈ mode for this symmetrical distribution, and 66.5% of frogs lie within one standard deviation of the mean.

Oct 3, 2026

Uploaded by Kurz