Week 1: Chapter 2 Data Representation

Tabular display of distribution

Frequency distribution

Cumulative frequency distributions

Relative frequency distributions

Cumulative relative frequency distributions

What questions does Frequency Distribution answer

-Which score occurred the most frequently?

-Which scores were the highest and lowest?

-Where do most of the scores tend to fall?

Symbol X meaning

The score/value

Symbol f meaning

Frequency, or number of times the score occurs

Symbol n meaning

sample size, or total number of observations.

Largest f. What score occurred most frequently? (aka Mode)

X=16 f=1

X=17 f=5

X=18 f=3

X=17 f=5

Cumulative Frequency

How many observations have that score OR a lower score

Cumulative frequency at 9

X=9 f=1

X=10 f=1

X=11 f=2

cf=1; 1 person scored 9 or lower

Cumulative frequency at 10

X=9 f=1

X=10 f=1

X=11 f=2

cf=2; 2 people scored 10 or lower

Cumulative frequency at 11

X=9 f=1

X=10 f=1

X=11 f=2

cf=4; 4 people scored 11 or lower

Cumulative frequency at 12

X=9 f=1 cf=1

X=10 f=1 cf=2

X=11 f=2 cf=4

X=12 f=1

(cf)+(f)=next cf

4+1=5

cf=5; 5 people scored 12 or lower

Relative Frequency

Percentage contained in an interval; aka a proportion or percentage

Relative Frequency Formula

f/n

Relative Frequency of 17

X=16 f=1 cf=12

X=17 f=5 cf=17

n=25

rf = f/n

5/25 = .20

=20%

Cumulative Relative Frequency

Percentage of scores in that interval and smaller

rf+crf

Cumulative Frequency at 9

X=9 f=1 cf=1 rf=.04

X=10 f=1 cf=2 rf=.04

X=11 f=2 cf=4 rf=.08

.04 → 4%

Cumulative Frequency at 10

X=9 f=1 cf=1 rf=.04

X=10 f=1 cf=2 rf=.04

X=11 f=2 cf=4 rf=.08

.04+.04=.08 (crf)

.08→8%

Cumulative Frequency at 11

X=9 f=1 cf=1 rf=.04

X=10 f=1 cf=2 rf=.04

X=11 f=2 cf=4 rf=.08

.08+.08=.16 (crf)

.16→16%

Real limits

Midpoint of each value, considering decimals

Lower Real Limit (LRL)

Upper Real Limit (URL)

Upper Real Limit (URL)

Decimal values above whole number

Upper Real Limit (URL) of 18

18.5

Lower Real Limit (LRL) of 18

17.5

Find the midpoint of Real Limits

add values and divide them by 2

Midpoint between 17 and 18

(17+18)/2 = 17.5 (LRL)

(18+19)/2 = 18.5 (URL)

Interval Width (w)

Difference between the Upper and Lower Real Limits of an interval

How to find Interval Width (w)

w = URL-LRL

Interval Width of 18

URL = 18.5, LRL = 17.5

w = 18.5 - 17.5

w = 1

Ungrouped Frequency Distribution

Each score gets its own row

Score=9, Frequency=1

Score=10, Frequency=1

Score=11, Frequency=2

Ungrouped Frequency Distribution Interval Width

w=1

Grouped Frequency Distribution

Combine scores

Interval= 9-10, f=2

Interval = 11-12, f=3

Interval = 13-14, f=3

Grouped Frequency Distribution Interval Width

Interval= 9-10, f=2

Interval = 11-12, f=3

Interval = 13-14, f=3

w=2

Why group data

Listing every single score is not helpful.

over a hundred rows of data could be grouped into intervals

Bar Graphs are appropriate for which data

Nominal data

Bar Graphs

Separate and don’t touch

Histograms

Separated but touch. Because the values have an underlying order/continuity (continuous)

Histograms are appropriate for which data

Ordinal, interval, and ratio data

Frequency Polygons

find the midpoint of each interval, plotting frequency, and connecting them

Frequency Polygons are appropriate for which data

ordinal, interval, and ratio

What does X represent in Frequency Polygon (X, Y)

Interval midpoint

What does Y represent in Frequency Polygon (X, Y)

Frequency

Cumulative Frequency Polygon

Continuously increases or stays the same, connected throughout and cannot decrease

Uses cumulative frequencies rather than regular frequencies.

Points are plotted at the upper real limits

What does X represent in Cumulative Frequency Polygon

scores/interval

What does Y represent in Cumulative Frequency Polygon

Cumulative Frequency

Shapes of Frequency Distribution: Normal

Symmetrical, bell-shaped

Most observations are around the center.

Fewer observations at both extremes.

The left and right sides are approximately mirror images.

Shapes of Frequency Distribution: Positive

Long tail points right

Most scores are low. A few scores stretch toward the high side.

Most scores are low, but 20 stretches the distribution toward the high end

Shapes of Frequency Distribution: Negative

A few low scores stretch toward the lower side.

Tail toward the negative/low side.

Mostly high scores and a few lower ones

Stem-and-Leaf

Lets you see the distribution while retaining the original scores.

Reconstruct original scores

In Stem-and-Leaf what does the Stem represent

Left vertical values

First part of the number/value

Stem Example in Stem-and-Leaf

4, 20, 83, 201

0

2

8

20

In Stem-and-Leaf what does the Leaf represent

Remaining digit/unit of each score

Ones, behind decimal, etc.

Leaf Example in Stem-and-Leaf

4, 20, 83, 201

4

0

3

1

Correct Stem-and-Leaf Display

4, 20, 83, 201

0|4, 2|0, 8|3, 20|1

Percentile

A score below which a certain percentage of the distribution falls.

Actual scores that are continuous values

Percentile example

50th percentile equals 150

50% of the distribution falls below a score of 150.

P50 = 150

Percentile Rank

Percentage of distribution of scores that fall below or less than a certain score

Percentile Rank example

Rank 150 has a percentile rank of 50

PR(150) = 50

Quartile ranks

Q1, Q2, Q3

Each rank has 25%

Quartile 1

Q1 = P25

25th Percentile

Quartile 2

Q2 = P50

50th Percentile = median

Quartile 3

Q3 = P75

75th Percentile

Percentile Calculation